ISA Transactions 39 (2000) 317±325 www.elsevier.com/locate/isatrans PID gain scheduling using fuzzy logic T.P. Blanchett a, G.C. Kember a,*, R. Dubay b a Department of Engineering Mathematics, DalTech, Dalhousie University, PO Box 1000, Halifax, NS, Canada B3J 2X4 b Department of Mechanical Engineering, University of New Brunswick, Federicton, NB, Canada E3B 5A3Abstract A simple, yet robust and stable alternative to proportional, integral, derivative (PID) gain scheduling is developedusing fuzzy logic. This fuzzy gain scheduling allows simple online duplication of PID control and the online improvementof PID control performance. The method is demonstrated with a physical model where PID control performance isimproved to levels comparable to model predictive control. The fuzzy formulation is uniquely characterized by; (i) onefuzzy input variable involving the PID manipulated variable, (ii) two parameters to be tuned, while previously tunedPID parameters are retained, and (iii) a gain scheduling dierential equation which relates the fuzzy and conventionalPID manipulated variables and enables fuzzy gain scheduling. # 2000 Elsevier Science Ltd. All rights reserved.Keywords: Gain scheduling; Fuzzy control; Model predictive control; PID control1. Introduction desired and predicted responses. However, chan- ging to MPC is not justi®ed for the majority of Most industrial process control continues to rely industrial PID controllers since its control struc-upon `classical, or `conventional proportional, tures are very dierent from PID, are much moreintegral, derivative (PID) control. Gain scheduling complicated, and have an increased computationalis the most common PID advancement used in cost.industry to overcome nonlinear process character- Fuzzy logic approaches have been shown inistics through the tailoring of controller gains over numerous studies to be a simpler alternative tolocal operating bands. This scheduling is compli- improve conventional PID control performancecated by the need for detailed process knowledge (for example, [1±5] for a recent overview). The pro-to de®ne operating bands and open loop tests which blem of interest here, is the control of a manipulatedmust be performed to locally calibrate the controller variable to a constant set point. Performancegain within each band. An alternative method is improvements for such a problem are usuallypredictive control which uses a `black box model to demonstrated by reductions in the amplitude ofremove the need for detailed knowledge of process undesirable oscillations in the manipulated vari-characteristics. For example, in model predictive able around the set point, shorter times to convergecontrol (MPC), controller moves are determined by to the set point, and the maintenance of controlcontinuously minimizing the dierence between the stability seen in conventional PID control. Since substantial, but similar improvements are found * Corresponding author. Tel.: +1-902-494-3262; fax: +1- from a wide variety of fuzzy logic schemes, the902-494-1801. main feature which delineates these approaches is E-mail address: email@example.com (G.C. Kember). their relative complexity. For those fuzzy logic0019-0578/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.PII: S0019-0578(00)00024-0
318 T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325controllers intended to replace existing conven- Note that a fuzzy logic scheme incorporatingtional PID controllers in the industrial setting, the these features is a true gain scheduler Ð a `fuzzydrive to simplify fuzzy logic controllers is impor- gain scheduler. Fuzzy gain scheduling is com-tant to reduce the costs of their implementation pactly and generally formulated in terms of a `gain. Two features shared by most of these fuzzy scheduling dierential equation: the rate oflogic setups are: each error component (taken change of the fuzzy manipulated variable is equa-from the proportional error and its derivatives) is ted to a function of the rate of change of the con-de®ned as a separate input, and the fuzzy rule- ventional PID manipulated variable. The form ofbases are redundant, that is, the rulebases show a this function is globally determined by details oflinear dependence upon the error components. the fuzzy formulation and the defuzzi®cationSuch `fuzzy redundancy together with appro- strategy. The existence of a limiting linear form ispriate input and output bounds has been shown to used to preserve conventional PID control andlead to stable control in a large class of nonlinear allow the desired online replacement. Then, mod-control problems . However, a practical obser- i®cation of this linear form, to a nonlinear sig-vation [6,7] is that fuzzy input variables taken moidal form, yields fuzzy gain scheduling. The usefrom linear combinations of the error components of a dierential equation also makes this approach(termed here `summed fuzzy input variables) should equally convenient for continuous and discretebe used to reduce the number of input variables control situations.where separated inputs would lead to a more The layout of the paper is as follows. The con-redundant rulebase. Such designs are simpler and trol of a temperature process by conventional PIDthus provide more ecient control than the more is used for illustration (Section 2). The fuzzy gainredundant fuzzy formulations, yet do not sacri®ce scheduling method and approach to independentstability . In addition, control robustness with tuning of parameters is developed (Section 3) andrespect to parameter ¯uctuations, seen in most demonstrated with a physical model (Section 4). Afuzzy designs is related to widespread use of error well-tuned PID controller is substantially improvedcomponents involving the proportional error and to performance levels of the benchmark MPCits derivatives , i.e. there is no integral term of after tuning the fuzzy gain scheduling method withthe error, and such control has been coined `slid- a few tests (Section 5). Excellent control robust-ing mode control in . ness and stability to large disturbances and large Therefore, the aim of this study is to provide a set point modi®cations is also demonstrated.new fuzzy formulation which provides a signi®cantsimpli®cation over existing fuzzy-PID schemesintended to improve conventional PID controllers. 2. PID controlThe larger simplicity of the method stems fromthree features: The control of a temperature process to a set point temperature is used to illustrate the fuzzy 1. Fuzzy redundancy is eliminated by using gain scheduling developed here. For the control of only one fuzzy input variable proportional to a temperature process by varying heater power, the derivative of the conventional PID the heater power is determined in conventional manipulated variable. PID control by manipulating 2. Online replacement and subsequent improve- ! ment of PID control is simpli®ed through the 1 t d À t Kp e t e udu Td e t Y I introduction of a dierential equation relat- Ti 0 dt ing the fuzzy input and output variables. 3. Online control improvement is achieved by the where the error at time t is e Ts À T; T is the independent tuning of only two parameters, process temperature, and Ts is the process set while the previously tuned conventional PID point temperature. The three PID control para- parameters Ti and Td are retained. meters are: the proportional gain Kp , the integral
T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 319time constant Ti , and the derivative time constant with initial condition g 0 2 0. Hence, gainTd . The heater power, P is equal to À, but P is set scheduling of the input, d2ad(, is modelled in (3)to 0 or the maximum heater power Pmx , when À is as a nonlinear dependence of the output dgad(respectively less than 0, or is greater than Pmx . upon d2ad(. The positive scaling constants and The temperature T is conveniently rescaled with are necessary to scale the fuzzy input and outputrespect to the set point temperature and the ambient respectively (this is further detailed in Section 3.3),temperature TI , using 0 T À TI a Ts À TI , and the dimensionless heater power, P, equals gso that TI 4T4Ts corresponds to 04041. The truncated to the range (0,1), i.e. P 1 when g b 1,time is also rescaled, using a timescale ts , as ( tats . and P 0 when g ` 0. Conventional PID controlWith these de®nitions, the dimensionless error is is generally recovered (`fuzzy logic equivalent)E 1 À 0, and if 2 ÀaPmx , then the dimen- when g 2; if f d2ad( d2ad( and ,sionless form of the manipulated variable (1) is then integration and application of the initial ! condition, g 0 2 0, yields g 2. Note that, Ã 1 ( Ã d although fuzzy gain scheduling could also be2 ( Kp E ( Ã E udu Td E ( X P Ti 0 d( based upon 2 instead of d2ad(, and this may seem attractive for 2 perturbed by noise, the tradeo is Now, P, the dimensionless heater power, is equal that it introduces an increased sensitivity to para- to 2 truncated to the range [0,1], i.e. P 1 when metric ¯uctuations. Hence, the approach taken2 b 1, and P 0 when 2 ` 0. The dimensionless here is to utilize the robustness associated withPID control parameters are KÃ Kp Ts À TI a p sliding mode control , and to supplement thisPmx Y TÃ Ti ats , and TÃ Td ats . i d with explicit signal processing for noise suppres- sion (Section 5). A discrete equivalence to PID is also important3. Fuzzy gain scheduling for discrete control applications, such as pulse width modulation (Section 4). Assume that the Fuzzy gain scheduling is in three steps: a fuzzy process is sampled at intervals of ts seconds sologic system is built that incorporates the features that the dimensionless sampling interval is unity.listed in the Introduction while preserving con- If d2ad( in (3) is approximated, at ( n, asventional PID control (Section 3.1), gain schedul- 2n À 2nÀ1 [note, any dierencing scheme producesing is then implemented by modifying this system a fuzzy logic equivalent if it is applied to both(Section 3.2), and two parameters are indepen- sides of (3)], and g at ( n is gn , thendently tuned (Section 3.3) to improve PID controlperformance. 1 gn gnÀ1 f 2n À 2nÀ1 X R 3.1. Fuzzy logic system The fuzzy input variable is taken to be equal to At ( n, the power P is Pn , and Pn is equal tothe rate of change of the PID manipulated vari- gn truncated to the range (0,1). The fuzzy logicable 2 in (2). Gain scheduling of d2ad( ensures equivalent now follows the continuous case.that control is less susceptible to parameter ¯uc- The parameter [(3) and (4)], is necessary totuations  since control near the set point always scale the input d2ad( to O 1 (`Ocorresponds to d2ad( 0 (sliding mode control means `the order of). Therefore, the function f,). Gain scheduling of d2ad( is formulated using that the fuzzy logic system must reproduce to a dierential equation where the rate of change of obtain a fuzzy logic equivalent is; f Y 41,the fuzzy output (manipulated) variable satis®es and it is further assumed that f 1, 51, and f À1Y 4 À 1. Given that f over 41,dg 1 d2 it is also clear that the fuzzy logic equivalence f Y Qd( d( relating to (3) and (4), requires such
320 T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 that d2ad( 41. Note that, in practice 2 has increase f, which are respectively denoted as Yd ,superimposed noisy perturbations and conditions Yn , and Yi and these sets are all unity respectively Â Ã Â Ã Â Ãdo change between control runs. Hence, a peak in f P À 3 Y À 1 Y f P À 1 Y 1 , and f P 1 Y 3 , and are 2 2 2 2 2 2value of d2ad( is normally estimated from pre- 0 otherwise. The consequence of each rule isvious control runs, and this value is used to deter- represented as a fuzzy set following . Formine . A fuzzy logic system that reproduces this example, the consequent for Rule 1, labelled asf is developed now. To implement fuzzy gain the set Y1 , is equated to the fuzzy set Yd , wherescheduling it is necessary to at least resolve the the maximum value of Y1 is Xn . Following thescalar inputs into three domains: negative, near same procedure for the three rules gives thezero and positive (this point is further examined in Â Ã three consequence sets: (i) Y1 Xn Y f P À 3 Y À 1 ,Section 3.2 where the generalization to more than Â Ã 2 2 (ii) Y2 Xz Y f P À 1 Y 1 , and (iii) Y3 Xp Ythree domains is also outlined). Therefore, three Â1 3Ã 2 2rules, relating the scalar input , and the scalar f P 2 Y 2 . It remains to evaluate the scalar outputoutput f, are introduced: Rule 1; IF is negative f. Adopting an additive centroidal defuzzi®cationTHEN decrease f, Rule 2; IF is zero THEN do strategy nothing to f, Rule 3; IF is positive THENincrease f. The three input fuzzy sets are negative, 3 Aj cjzero, and positive, and the three output fuzzy sets j1are, decrease f, do nothing to f, and increase f. It is f X S 3necessary to convert these three rules into a com- A j j1putational framework. This requires a means; (i)to compute the degree of membership of the scalarinput in the input fuzzy sets, or the IF portion of Aj Y j 1Y 2Y 3, are the areas respectively corre-each rule, (ii) to evaluate the consequence of sponding to the consequent fuzzy sets Yj Ymembership in each set, or the THEN portion of j 1Y 2Y 3; A1 Xn , A2 Xz , A3 Xp .each rule, and (iii) to estimate the scalar output f The values c1 À1, c2 0, c3 1 are the respec-from the three consequences of membership eval- tive centroids of these consequent sets. The sum ofuated in (ii). the areas in the denominator of (5) is always unity The input fuzzy sets, negative, zero, and posi- since the components x sum to unity. The tive, are respectively denoted as Xn Y Xz Y Xp . The numerator evaluates to for 41, and is 1 fortypical linear sets are used, i.e. Xn ÀY and À 1 for 4 À 1. Hence f for 51À1440 Xn 1Y 4 À 1, and 0 otherwise), 41, while f 1Y 51, and f À1Y Xz 1 À Y 41 (0 otherwise), Xp Y 04 À 4 À 1, that is, f preserves conventional PID.41 Xp 151, and 0 otherwise). The degree ofmembership of the scalar input in the input 3.2. Gain schedulingfuzzy sets, is evaluated as the three scalars:Xn Y Xz , and Xp , and these are stored in Â Ã Global PID control performance can bethe vector x Xn Y Xz Y Xp . When 41, improved by scheduling the gain KÃ as a function pthe control is not truncated, and x ÀY 1 of the derivative of the PID manipulated variableY 0Y À1440, and x 0Y 1 À Y Y 0441. d2ad(. More speci®cally, the sensitivity to smallSimilarly, when 51, the control is truncated, and deviations from the set point is increased, andx 1Y 0Y 0Y 4 À 1, and x 0Y 0Y 1Y 51. The the reverse is applied to larger deviations, i.e.locations where the control is truncated are chosen dfad is increased near 0, and decreased near without loss of generality as 1 and À1, 1, so that f is sigmoidal. This is achievedsince the inputs are scaled to ensure is order 1. here by applying variable weights to the con- To evaluate the consequence of membership, it sequent fuzzy sets ( does so for an unrelatedis necessary ®rst to de®ne the output fuzzy sets, i.e. problem) so that the defuzzi®cation strategy in (5)the three fuzzy sets decrease f, do nothing to f, becomes
T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 321 3 right-hand side of (8) is a monotone function of wj Aj cj the derivative of the PID manipulated variable, j1f X T absolutely bounded by 1a (this is analogous to 3 wj Aj the statement regarding stability made in  noted j1 in the Introduction). Note, for control problems where the gain scheduling requires ®ner control of The weights are positive, and conventional PID the sigmoidal shape of f , the number of setsis recovered in the same fashion as in Section 3.1 may be increased and this simply adds extrawith the additional requirement w1 w2 w3 . weights to the defuzzi®cation strategy.Since the form of (6) is unchanged when theweights are multiplied by a constant, the weight w2 3.3. Parameter tuningis set to unity without loss of generality, andattention is further restricted to symmetric weights There are three parameters in the fuzzy gainw1 w3 w. Applying both of these to (6) gives scheduling in (8): , , and w. The parameter is for 41 used to scale d2ad( to order 1, so that control far from the set point is d2ad( O 1, and w near to the set point is d2ad( ( 1. More pre-f Y U w À 1 1 cisely, the necessity of preserving conventional PID control is used to ®x ; if is such that where f 1 for b 1, and f À1 for ` À1. The d2ad( 41, where d2ad( is taken from theend point values of f Æ1 Æ1, and f 0 0 are existing PID manipulated variable, then the fuzzyindependent of w, in contrast to the slopes logic equivalent follows from , and w 1.dfad 0 w, and dfad Æ1 1awY f is sig- Therefore, it is only necessary to tune two para-moidal when w b 1 and this is the desired gain meters, and w to globally improve the existingscheduling described above. The derivative of f PID control performance. A key observationis also continuous at 0 so that special treat- leading to independent tuning of and w is thatment of control near, and across 0  is avoi- for improvement of well-tuned PID, O .ded and this justi®es the restriction to symmetric Then dgad( O wd2ad( near the set point andweights. Furthermore, the parameter 3 de®nes the control sensitivity near there is O w. Therefore,extent to which inputs near 0 in¯uence the output control sensitivity near the set point is increased byrelative to those further away from 0 and thus it is setting equal to and independently tuning w bnecessary to de®ne at least three sets (as in Section 1 to reduce maximum set point overshoot. Next, 3.1) since inputs can at least be, near zero, large is independently modi®ed to b to reduceand positive, or large and negative. control sensitivity far from the set point and fur- Substituting f (3) gives explicitly for ther reduce maximum set point overshoot. Whilst,d2ad( 41 is varied, the sensitivity near the set point is maintained at the previously tuned w, by modifyingdg 1 w d2ad( w such that wa is unchanged. A physical model Y V (Section 4) is now used to demonstrate (Section 5)d( w À 1d2ad( 1 improvement of well-tuned PID control.where dgad( 1a for d2ad( b 1, and dgad( À1a for d2ad( ` À1. To recover conventional 4. Physical modelPID control; 3 1, , and is chosen such that d2ad( 41, whereupon (8) reduces to The control of a temperature process, depicteddgad( d2ad(, and then integration and appli- in the schematic in Fig. 1, is conducted on a solidcation of g 0 2 0 yields the desired g 2. cylindrical block of aluminum, 5 cm diameter andControl based upon (8) is stable, since the 12.5 cm in length. The block is externally heated
322 T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 Fig. 1. Experimental setup for control of temperature process by pulse width modulation.by a 300 watt electrical heater band wrapped power setting, a continuous variable, it is easilyaround the block circumference. A type E, modi®ed to the discrete pulse width modulation.ungrounded thermocouple, measures the objects To avoid confusion, the nomenclature in Sectionstemperature at its center, and these analog mea- 2 and 3.1 is adopted. During the nth duty cycle,surements are converted to digital readings using a the on time of the heater, or pulse width Pn s, is12 bit analog-to-digital converter. Process control determined by the control algorithm, while theis over contiguous duty cycles of constant dura- heater power setting is held ®xed between dutytion. The heater is on for a portion of a duty cycle, cycles. The maximum pulse width, Pmx s, is equalstarting at the beginning, and then o for the to the duty cycle duration. The average errorremainder; the heater on time during a duty cycle within a duty cycle is en Ts À Tn where Tn is theis termed the pulse width. A pulse is implemented average temperature over a duty cycle. The timeusing a 16 bit digital timing board, and an opti- scale, for the dimensionless form, is taken to becally isolated solid state SSR-20 electronic relay. the duty cycle duration (the sampling interval ofTwo logic states, on and o, corresponding to the the average temperature), and the nth duty cycle isheater being on or o, are generated by the digital then over n À 14(4n. The average dimensionlesscounter and are inputted to the relay. The process error within the nth duty cycle, is En 1 À 0n ,control algorithm determines the duration of the where 0n Tn À TI a Ts À TI . Finally, Pn Y 2n ,on logic state for each duty cycle, or modulates the and gn , follow the description in Section 3.1.pulse width between duty cycles Ð hence pulse The general approach followed here to ®lterwidth modulation. The duty cycle duration is noisy ¯uctuations from the error components (i.e.empirically set at 4.25 s, and at steady state this the error variable and its derivatives), does notcorresponds to a maximum error, over a duty rely upon features of the control setup or choice ofcycle, of less than 1% (the thermocouple accuracy sampling period (prone to aliasing errors). Rather,is about 1%). the error variable is ®rst sampled at a high enough rate to establish all features relevant for the con- trol application. Then, each independent error5. Results component is separately processed for noise sup- pression. For the experiments conducted here, the Although the process control described in Sec- average temperature Tn during a duty cycle is thetion 2 is based on the determination of heater average of 10, equally spaced temperature mea-
T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 323surements. A least squares regression is also used and the fuzzy logic equivalent corresponds to to reduce noise in the error and the numerical 38 and w 1. Fuzzy gain scheduling is usedapproximation of its derivatives. Speci®cally, the to improve upon the existing PID control perfor-error value En (essentially Tn ) and its ®rst deriva- mance by tuning the parameters and w awaytive are calculated from a line, and the second from and w 1. Firstly, maximum set pointderivative from a quadratic polynomial, all least overshoot is reduced by increasing control sensi-squares regressed on measurements taken from the tivity near the set point. This is achieved by inde-most recent 16 duty cycles. The choice of 16 duty pendently tuning 3 (Section 3.3) with fourcycles (about 1 min) is arbitrary, but is chosen to experiments w 2Y 4Y 6Y 8, where 38. Thebe much smaller than the process time constant value w 6 is chosen (see Fig. 2; w 2Y 4Y 8 are(about 500 duty cycles or 30 min). The least not shown) since it gives about a ®vefold reduc-squares regression is eciently implemented as a tion in maximum overshoot and w 8 providesconvolution using the Savitzky±Golay formula- marginal additional improvement. The maximumtion . The set point temperature is chosen as overshoot is somewhat reduced again, by decreas-Ts=100 C and the ambient temperature is ing the control response, dgad(, away from the setapproximately 25 C. The dimensionless PID con- point and this corresponds to b . The pre-trol parameters KpÃ 1Y TiÃ 37X5, and TdÃ 9X4, viously tuned control sensitivity near the set pointand the temperature response 0 is shown in Fig. 2 is maintained at w 6 by varying w such that wa[well-tuned PID control ( 38, w 1) in the is constant, while is increased by 20, 30, and®gure] as a function of the dimensionless time (. 40%. The overshoot for each of these values isThis control (tuned for minimum overshoot) shows about 3, 2, and 10%, respectively. Hence, as isapproximately one quarter amplitude damping increased the maximum overshoot is ®rst reducedwith settling time approximately the process time by the initial reduction in control sensitivity farconstant and is typical of well-tuned PID. This from the set point, but then increases as the controltemperature response can be greatly improved by sensitivity becomes too reduced. A 30% increase overfuzzy gain scheduling. is chosen to scale 38 is chosen and the ®nal results are shown ind2ad( to order 1. From the existing PID Fig. 2 for 50 which also corresponds to w 7X8manipulated variable (not shown), if % 38, then (a 30% increase over 3 6). The maximum over-d2ad( 41 for the duration of the PID control, shoot has now been reduced about sixfold to 2.5% maximum overshoot. Five indices are also used to assess the overall control performance: the max- imum overshoot and undershoot of the tempera- ture expressed as a percentage of the set point, the rise time, which is the time needed to rise to within 90% of the set point, the settling time, or time the process requires to fall within Æ2.5% of the set point, and the steady state error. These ®ve indices are presented for the conventional PID control w 1Y 38 fuzzy gain scheduled control w 6Y 38Y nd Y w 7X8Y 50, and MPC control in Table 1. From Table 1 and Fig. 2, it is clear that fuzzy gain scheduling ( 38Y w 7X8, and 50) provides much better control perfor- mance than the well-tuned PID control w 1Y 38. In particular, the settling time isFig. 2. Temperature response for conventional PID control, reduced to about one half the process time con-fuzzy gain scheduled PID, and model predictive control stant, and the percentage maximum overshoot is(MPC). reduced from 15 to 2.5%.
324 T.P. Blanchett et al. / ISA Transactions 39 (2000) 317±325 The benchmark experiment is based on MPC. In in 50 cm3 of water at 10 C for 5 s), and largethe MPC approach used here (details are in ) a (40 C) changes in the set point (Fig. 4), is clearlydiscrete step response of the physical model is demonstrated for the tuned fuzzy gain schedulingobtained by an open loop test. The method utilizes 38Y w 7X8Y 50 where the manipulatedtwo horizons: a `control horizon equal to thenumber of predicted control moves, and a `predic-tion horizon equal to the number of samplingintervals to reach 95% of the open loop steadystate. Predictions of the physical model output aremade within the prediction horizon, and these arecompared to the desired set point pro®le. Leastsquares minimization of the dierence between thepredictions and the set point pro®le, over the pre-diction horizon, is used to determine the manipu-lated variable within the control horizon. Acontrol horizon of length 2 and a prediction horizonof length 139 was used here for controlling thetemperature. Although MPC control is funda-mentally dierent from conventional PID, it pro-duces control actions similar to PID control, butshows a very reduced overshoot and settling time tothe set point due to its predictive capability. Thus, Fig. 3. Temperature response of fuzzy gain scheduled PID to aMPC is practically useful to provide a range of disturbance (applied at ( % 280), and the same for model pre-comparison to fuzzy gain scheduling. A surprising dictive control (MPC) (applied at ( % 350). The fuzzy gain scheduled response is almost Identical to that seen for MPC.result, evident in Fig. 2, is that the fuzzy gain sche-duling fares very well in comparison to the moresophisticated benchmark MPC control. Better per-formance indices were also obtained for gainscheduling, w 7X8Y 50, over MPC controlwhen only the rise time and settling time are con-sidered, and marginally worse results for percen-tage maximum overshoot and undershoot. Control robustness to a short, cooling disturbance(Fig. 3) (the cylindrical block was suddenly placedTable 1Control performance indicesPerformance indices w 1, w 6, w 7X8, MPC 38, 38 38 38 38 38Percentage 15 3.2 2.5 0 maximum overshootPercentage 4 1 1 0 maximum undershootRise time (min) 7.7 7.3 6.9 14 Fig. 4. Temperature response (a) and manipulated variable (b)Setting time (min) 30 15 14 19 for fuzzy gain scheduled PID. Control is depicted for set pointsPercentage Æ0.5 Æ0.5 Æ0.5 Æ0.5 40% larger Ts 140 g and smaller Ts 60 g than that steady state error used for the parameter tuning Ts 100 g.
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