silicon wafers p b θ b IR heater θ h p h solution bath cleaning filter bellows pump

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1 An Analysis of Cometitive Assoiative Net for Temerature Control of RCA Cleaning Solutions S Kurogi y, H Nobutomo y, T Nishida y, H Sakamoto y, Y Fuhikawa y, M Mimata z and K Itoh z ykyushu Institute of Tehnology, Kitakyushu 84{855, Jaan kuro@ntlkyutehaj zkomatsu Eletronis In, Hiratsuka 254{8543, Jaan Abstrat This aer illustrates and analyzes the ometitive assoiative net alled CAN2 alied to modelswithing ontrol of the temerature of RCA leaning solutions, where the RCA lean is the industry standard way to lean silion wafers and the temerature ontrol is imortant for a stable leaning erformane Sine the ontrol is diult owing that the RCA leaning solutions exose exothermi hemial reations being nonlinear and time-varying, we have resented an arhiteture of model-swithing reditive ontroller (MSPC) using CAN2, so far Here, we analyze the behavior of the MSPC in detail and show the advantages of the arhiteture, suh as the model-swithing sheme omensating slow learning seed, the inremental learning ability of the CAN2, et by means of omuter simulation 1 Introdution The ometitive assoiative net alled CAN2 has been roosed for funtion aroximation and ontrol roblems [1, 2], and then alied to modelswithing ontrollers [3, 4, 5] Here, we would like to mention that the ability of the CAN2 in funtion aroximation has been shown eetive in the aliation to rainfall estimation, ie, the estimation using the CAN2 was awarded the 2nd rize in the rainfall estimation ontest held by Neuroomuting Tehnial Grou of IEICE (Institute of Eletronis, Information and Communiation Engineers) of Jaan[6] On the other hand, the RCA lean, whih was develoed at Radio Cooeration of Ameria, is the industry standard way to lean silion wafers, where the temerature ontrol of hemial solutions is imortant for a stable leaning erformane Sine the RCA leaning roedure uses orrosive and hazardous hemial solutions suh as SPM (sulfuri aid H 2 SO 4 and hydrogen eroxide H 2 O 2 mix- This work was suorted by the Grant-in-Aid for Sienti Researh (no ) of the Jaanese Ministry of Eduation, Siene, Sorts and Culture IR heater θ h h bellows um leaning filter silion wafers b θ B θ b solution bath θ Figure 1: Shemati diagram of RCA leaning system, where reresents the inut ower and B the outut temerature ture) and so on, several seial equiments are arranged for heating solutions as shown in Figure 1, where there are a solution bath, a bellows um, an infrared (IR) heater and a leaning lter whih are onneted by anti-orrosive reirulation ies, and the thermal sensor is overed by an anti-orrosive glass tube Thus, this system involves long and utuating time lags and delays, and the mixture of the solutions exose several exothermi reations whih are nonlinear and time-varying In order to ontrol suh a nonlinear and timevarying dynamial system, we have introdued the model-swithing reditive ontroller (MSPC) using CAN2 [4] The CAN2 basially learns and ahieves ieewise linear aroximation of nonlinear funtion, while it rovides a linear model (the arameter values of a linear funtion) to the MSPC from time to time in order to ontrol nonlinear and time-varying lants 2 CAN2 and MSPC Here, we illustrate the CAN2 and the MSPC, where we have several versions of CAN2 and the

2 u(k-1) lant y Figure 2: Disrete model of the system x M 1 1 y(k-1) y(k-n y ) u(k-1) u(k-n u ) C 1 M C M N C N y 1 y y N C y ^ = y k-ne =argmin Σ i j=k-1 [y(j)-y i(j)] 2 Figure 3: Shemati diagram of the CAN2 for aroximating y = f(x) + d resent one is alled simlied CAN2 in [4] and CAN2-2 in [5] 21 Disrete model of the system First, let us onsider a disrete model of the system or the lant asshown in Figure 2, whose inut u and outut y at disrete time k =; 1; 2; has the relation given by y =f(x) + d; (1) where the funtion f does not hange or hanges slowly in time, d reresents noise, and the vetor x 2 R n1 is given by x =(1;y(k, 1); ;y(k, n y ); u(k, 1); ;u(k, n u )) T ; (2) where n y and n u are nonnegative integer onstants and n =1+n y + n u 22 CAN2 The CAN2 onsists of several ells C i for i 2 I = f1; 2; ;Ng (see Figure 3 for shemati diagram) Eah C i has an assoiative matrix M i 2 R 1n for aroximating y in Eq(1) by y i =M i x; (3) and the CAN2 selets to outut one of them as by =y =M x; (4) where M is alled seleted or identied model, and the seletion of the index is done by the ometition through the following minimization roess = = argmin i2i k,ne X j=k,1 y(j), M i x(j) 2 : (5) 23 Learning algorithm The learning algorithm is to minimize the sum of squared aroximation errors of the net E = 1 k kx j=1 y(j), by(j) 2 : (6) whih is the funtion of x i and M i (i 2 I) being otimized by the following roedure Ste 1: (Cometition) Searh the th ell by (5) Ste 2: Ste 3: (Re-initialization) If y, by e and v v, then searh the lth ell with the minimum ring number v l of all v i (i 2 I), and then set M l := M and v l := v, where v and e are thresholds, and := indiates substitution (Assoiative learning) There are several versions of udating assoiative memory M ; the simlest one is M := M +(y, M x)x with a small ositive onstant, a strit one uses M, and [5], and we here use the following inremental algorithm for least square solution (see [4] for details); M := M + (y, M x) x T 1+x T x ; (7) :=, xx T ; 1+x T x (8) v := v +1; (9) where y = y, x = x, and 2 R nn has the initial value = ai nn with a large number a and the unit matrix I nn See Set4 for atual values of a and the initial M used for the ontroller 24 MSPC At time k, the aroximation of the lant dynamis is given by by = M x, and then, the redited inut bu(k + j) (j =; 1; ) for minimizing the ost funtion J = XN 2 j=n 1 be 2 (k + j)+ u XNu j=1 bu 2 (k + j, 1); (1) an be alulated and set as the lant inut u = bu, where be(k + j) = y d (k + j), by(k + j) is the redited error between the desired outut y d (k + j) and the reditive outut by(k + j) whih is the resonse to bu(k + j, i), and bu(k + j, 1) = bu(k + j, 1), bu(k + j, 2) is the ontrol inrements The arameters N 1, N 2, N u and u are onstants to be designed From the GPC (generalized reditive ontrol) method [7], we have J = ky d, Gbu, y k 2 + u kbuk 2 ; (11)

3 where z(kt) y d =(y d (k + N 1 ); ;y d (k + N 2 )) T (12) y =(y (k + N 1 ); ;y (k + N 2 )) T (13) z(kt-t v ) bu =(bu; ;bu(k + N u, 1)) T ; (14) and y (k + j) for j = 1; 2; is the resonse of the system assuming that u(k + j) =u(k, 1) (j = ; 1; ) The element at the ith olumn and the jth raw of the matrix G 2 R (N 2,N 1 +1)Nu is G ij = g i,j+n1, and g k (k = ;,2;,1; ; 1; 2; ) is the ste resonse of the system, y, for y(i) = u(i) = (i < ) and u(i) = 1 (i ) Then, the bu minimizing J is given by z(kt-2t v ) it (L-2)T v z L-2 T v it (L-1)T v z L-1 T v it LT v kt bu =(G T G + u I),1 G T (y d, y ); (15) and the ontrol is given by u = u(k,1)+bu 25 Virtual samling In order mainly to deal with long time lags and delays, we have introdued the virtual samling method for digitizing B (t) and (t) of the RCA leaning system, y(k, j) = B (kt, jt v ); (16) u(k, j) = (kt, jt v ); (17) where T is the real samling eriod, T v = K v T indiates the virtual samling eriod for a ositive integer K v, and B of B and of are dened as z of z given by 1 z(kt, jt v )= 2 + i z L,j K v + 1 2, i K v z L,j,1 (18) where z L is the average value of z(t) from t = LK v to t =(L +1)K v (see Figure 4) After all, the overall ontrol system is deited by Figure 5 3 Thermal Model of the Solutions We here briey exlain the thermal model [4] of the RCA leaning system; the outut themerature B for the inut ower is alulated with the following equations (see Figure 1 for shemati diagram): V h C d h dt =(, h + b (t, t b ))qc + + h (19) V b C d b dt =(, b + h (t, t h ))qc B d B dt +( O, b ) O V b C + b (2) =, B + b (t, t B ); (21) where h and b reresent the temerature at the heater and the bath, resetively, and V h, V b,, C, O, B, t h, t b and t B are onstants The variables Figure 4: Virtual samling method The dotted urve shows z(t) and the heavy solid line segment indiates the linear aroximation z(t)ofz(t) using the moving average values z L,1 and z L,2 y d (k+j) MSPC GPC: minimize J=Σ N2 Nu ^ j=1 e2 (k+j)+λ u Σ u^ 2 (k+j-1) j=1 M CAN2: learn and swith M i (i=1,2,,n) Virtual Samling: y(k-j)=θ B (kt-jt v ) u(k-j)=(kt-jt v ) u y (kt) RCA leaning solution θ B (kt) Figure 5: Shemati diagram of the MSPC for ontrolling the temerature of RCA leaning solution h and b reresent the exothermal ower given as follows; h = K( h )(1, h ) r (V h + L h S)H ; (22) b = K( b )(1, b ) r (V b + L b S)H : (23) where V h, V b, L h, L b and S are onstants, K() = ex(,e=r), and E and R are onstants The variable h and b are the arameters reresenting the ratio of the mass of reated exothermal substanes at the heater and the bath, resetively, and they hange from to 1 as follows; d h dt d b dt = K( h )(1, h ) r + ( h, b )q V h + L h S ; (24) = K( b )(1, b ) r + ( b, h )q V b + L b S : (25) 4 Simulation and Analysis We alied the MSPC to the above thermal model for the SPM So far, we have already examined and shown some results in [4], but here we

4 Figure 6: Result of the ontrol by the modied PID ontroller with K P = :11, K I = :46, and K D =:11 The overshoot is overshoot =2:37 C, and the settling time to C from the initial time t = 454s is t settle = 2395s Figure 7: Result of the ontrol by the MSPC with N = 1 The overshoot is overshoot = 2:69 C, and the settling time is t settle = 197s, and the assoiative matrix after the ontrol is M 1 = (+6:489; +1:718;,:768; +:5) examine harder ase where the initial temerature B = 12 C should be raised to the set oint 135 C with allowable error 2 C, and the onventional modied PID ontroller (see [4] for details) ould not ontrol the overshoot less than 2 C as shown in Figure 6, where b reresents the erentage mass of the reated exothermal substanes and the nonlinar and time-varying hemial reation aears before it reahes 1[%] (see Eq(22) to (25)) We rst tried only one ell (N = 1) for the MSPC and Figure 7 shows the result, where the overshoot is overshoot = 2:69 and this result is obtained after three iterations of 5,s duration of the ontrol for the online learning of the CAN2, and the overshoot ould not be redued even when the number of iterations was inreased Here, we have to mention that we set the arameter values for the MSPC as follows (see [4] for a detailed exlanation): for the CAN2, n y = 2, n u = 1, N e = 2, e = :1, v = 1, the initial values of the ells v i =, M i =(:; :99; :; :2) and i =1; I nn for all i; for the GPC, N 1 =, N 2 = N u = 4, and u =:2 The real and virtual samling eriods are T =:25s and T v = 8s, resetively Next, we tried ten ells (N = 1) and obtained the result shown in Figure 8, where the overshoot is redued to 1:2 C and then the settling time to C from the initial time t = 454s is t settle = 515s We an see that the seleted th ell hanges very frequently during b < 1, whih results in reduing the overshoot So, from Figure 7 and 8, we an say that the learning seed of the assoiative matrix given by Eq(7) or the CAN2 with N = 1 is too slow to adat the arameter hange owing to the exothermal reations, but the modelswithing sheme or the CAN2 with N =1ould omensate the slow learning seed by swithing the model or the assoiative matrix Now, let us examine the obtained assoiative matries M i = (M i ;M i1 ;M i2 ;M i3 ) and the seleted ones M ; let us take alook at the inut and the outut B from t = 4; to t = 5; in Figure 7 and Figure 8, where B are ontrolled very well for both ase, but the values of the seleted weight vetors M are dierent, where note that M = M i = M 1 in Figure 7 and the index = (t) of M in Figure 8(a) hanges and is seleted from i = 1; 2; ; 1 of M i in Figure 8(b) The reason why the dierent M an be used is as follows; the seleted matrix M indiates B (k +1) = M +M 1 B +M 2 B (k,1)+m 3, and the steady state for a onstant set oint derives that B (k +1)= B = B (k, 1) = B and = are onstant So, there are many M whih satisfy B = M + M 1 B + M 2 B + M 3 u The above analysis indiates that the obtained assoiative matries deend on the trajetory of the ontrol as well as the initial temerature So, the obtained matries M i for the initial temerature B = 12 C may not be useful for the initial temerature suh as B = 8 C even if it is an easier ontrol in general Thus, we did the simulation and obtained Figure 9, where we observe the outut B being well-ontrolled and the inut involving osillation or so-alled hunting Sine the hunting is not desirable, we made three more iterations of 5,s duration of the ontrol with the initial temerature B =8 C for online learning of M i And

5 (a) (b) i M i M i1 M i2 M i , , , , , , , , , , , , , ,78 5 Figure 8: Result of the ontrol by the MSPC with N = 1 (a) shows the resonse, where the dotted line indiates the seleted ell (= 1; 2; ; 1) multilied by 1 (for visualization) The overshoot is overshoot = 1:2 C, the settling time is t settle = 515s (b) shows the assoiative matries after the ontrol Figure 9: Result of the ontrol by the MSPC used in Figure 8 The overshoot is overshoot =:77 C, and the settling time is t settle = 1718s then, we obtained the result shown in Figure 1 where the hunting is disaeared Finally, we heked the ontrol with the MSPC used in Figure 1 for the initial temerature B = 12 C again, and obtained Figure 11, whih shows a good ontrol The above result indiates the so-alled inremental learning ability of the CAN2 as an advantage of the MSPC using the CAN2 Although it is hard to nd out how the matries M i in Figure 11(b) share the roles for the exothermal reations, for the initail temeratures B = 8 C and B = 12 C, resetively, the CAN2 atually uses the inremental learning algorithm given by Eq(7) for the assoiative matries to obtain least square solution of Eq(6) where E to be minimized is de- ned as the sum of squared errors from the initial time (see [4] and [8] for details) 5 Conlusion We have illustrated the ometitive assoiative net (CAN2) and the model-swithing ontroller (MSPC) for the temerature ontrol of RCA leaning solutions By means of omuter simulation, we have laried the behavior of the MSPC in ontrolling the temerature of RCA solutions, and show some advantages of the resent arhiteture, suh as the model-swithing sheme for omensating slow learning seed, and the inremental learning ability of the CAN2 For ratial aliations, we have to larify how many ells and how many learning iterations are neessary for various RCA solutions and various ontrol onditions, whih will be studied in our future researh Referenes [1] SKurogi and SRen, \Cometitive assoiative networks for funtion aroximation and ontrol of lants," Pro of NOLTA'97, 775{778, 1997 [2] SKurogi, TSakamoto and SRen, \An analysis of ometitive assoiative nets," Pro of ICONIP'98, 683{686, 1998 [3] SKurogi, HHirata, TNishida and TSakamoto Model-swithing adative ontrol using ometitive assoiative nets, Pro of NOLTA'99, vol2, 755{758, 1999 [4] SKurogi, TNishida, TSakamoto, KItoh and MMimata, \A simlied ometitive assoiative net and a model-swithing reditive ontroller for temerature ontrol of hemial solutions," Pro of ICONIP-2, 791{796, 2 [5] SKurogi and TNishida, \Adative and reditive ontrol using ometitive assoiative net for learning and swithing of multile models,"

6 Trans of SICE of JAPAN (in Jaanese),vol37, no3, 23{212, 21 [6] SKurogi, MTou and STerada, \Rainfall estimation using ometitive assoiative net," Pro of 21 IEICE General Conferene (in Jaanese), volsd-1, 26{261, 21 [7] D W Clarke and C Mohtadi, \Proerties of generalized reditive ontrol," Automatia, vol25, no6, 859{875, 1989 [8] TKohonen, \Assoiative Memory, A systemtheoretial aroah," Sringer-Verlag, 1977 Figure 1: Result of the ontrol by the MSPC after 3 more iterations of online learning for the initial temerature B = 8 C The overshoot is overshoot =:76 C, the settling time is t settle = 1695s (a) (b) i M i M i1 M i2 M i3 1, , ,2 1654, , , , , , , , , , , , , , , , , Figure 11: Result of the ontrol by the MSPC used in Figure 1 (a) shows the resonse, where the overshoot is overshoot =:52 C, the settling time is t settle = 583s (b) shows the assoiative matries after the ontrol