OBSERVER DESIGN FOR STATE ESTIMATION OF UV CURING PROCESSES

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1 Proeedings of the ASME 214 Dynami Systems and Control Conferene DSCC214 Otober 22-24, 214, San Antonio, TX, USA DSCC OBSERVER DESIGN FOR STATE ESTIMATION OF UV CURING PROCESSES Adamu Yebi Clemson University - International Center for Automotive Researh Greenville, SC 2967 ayebi@lemson.edu Beshah Ayalew Clemson University-International Center for Automotive Researh Greenville, SC 2967 beshah@lemson.edu Satadru Dey Clemson University - International Center for Automotive Researh Greenville, SC 2967 satadrd@lemson.edu ABSTRACT This artile disusses the hallenges of non-intrusive state measurement for the purposes of online monitoring and ontrol of Ultraviolet (UV) uring proesses. It then proposes a twostep observer design sheme involving the estimation of distributed temperature from boundary sensing asaded with nonlinear ure state observers. For the temperature observer, bakstepping tehniques are applied to derive the observer partial differential equations along with the gain kernels. For subsequent ure state estimation, a nonlinear observer is derived along with analysis of its onvergene harateristis. While illustrative simulation results are inluded for a omposite laminate uring appliation, it is apparent that the approah an also be adopted for other UV proessing appliations in advaned manufaturing. I. INTRODUCTION Ultraviolet (UV) radiation uring of materials is a widely used proess in photopolymerization of resins, oatings and adhesives. It is gaining substantial interest in other advaned manufaturing appliations suh as stereolithography and uring of omposite laminates due to its several advantages suh as 1) higher energy effiieny; 2) less environmental pollutions; 3) aelerated proessing time; 4) redued spae usage and maintenane osts, and; 5) better ontrollability[1-3]. Despite these advantages, the thikness of parts that an be ured effetively by UV-radiation is limited beause UV is attenuated as it passes through target materials aording to the so-alled Beer-Lambert effet [2]. As a result, extended irradiation may be needed to ure thik setions. However, in thik setions, the aompanying thermal and ure level gradients from the distributed exothermi ure reations may ompromise the quality and mehanial performane of the end produt. This is often overome by exhaustive offline optimization of proess parameters suh as exposure distane and intensity settings. For online monitoring and feedbak proess ontrol, online adjustment of the radiative inputs is desirable but the distributed ure state information throughout the thikness of the setion is rarely available in pratie. Most available ure level measurement options involve offline tehniques that an be done on laboratory samples destrutively. These inlude dynami mehanial analyzer (DMA), rheometers and thermal differential sanning alorimeter (DSC) [4]. In some appliations, there is the possibility of using online, albeit intrusive, measurements with non-reusable sensors (i.e. dieletri analyzer (DEA), fiberoptis, et.) that are imbedded in the material to be ured and beome part of the produt at the end of uring. The latter are often expensive and may affet the quality of the end produt as well[5]. In this paper, the design of state estimators/observers is onsidered as an eonomial, online, non-intrusive and nondestrutive alternative to overome the lak of suitable sensing options with these features for UV uring appliations. Manufaturing proesses involving uring suh as rapid prototyping, stereolithography[6] and layered omposite manufaturing[7] involve exothermi hemial reations ativated by thermal or radiative energy soures. In these proesses, the heat transfer in the part is modeled by a nonlinear partial differential equation (PDE) desribing the spatio-temporal evolution of temperature in the part and the ure kinetis is often modeled by a nonlinear ordinary differential equation (ODE)[6, 7] desribing the temporal evolution of the ure state. However, by virtue of the dependene of loal reation rates on loal temperatures, the 1 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

2 ure state is also spatially distributed and oupled to the temperature evolution. Despite this distribution of the temperature and ure states, the only non-intrusive pratial measurements are often limited to boundary measurement of surfae temperature via some optial sensors (IR thermal ameras). This limited sensing and the nature of the oupled nonlinear PDE-ODE models of the uring proess make the estimator/observer design problems hallenging. The literature offers a number of nonlinear observer design methods that have been tested for various finite dimensional systems modeled by ODEs: for example, Lyapunov-based methods(thau s[8] & Raghavan s[9]), geometri observer[1], and sliding mode observer[11]. Most of these are designed to guarantee onvergene of state estimation errors under deterministi model assumptions. For nonlinear stohasti systems, extended Kalman filter (EKF)[12] and unsented Kalman filter (UKF)[13] are used dominantly, although the stated theoretial onditions for error onvergene are often diffiult to establish/verify. In the ase of nonlinear infinite dimensional systems desribed by nonlinear PDEs, most of the observer designs extend the above methods by approximating the PDEs with representative ODEs. The ommon approximation tehniques are finite differene methods [14], or Proper Orthogonal Deomposition (POD)-Galerkin methods[15] While there appears to be limited prior work on observer designs speifi to UV uring appliations, some solutions have been proposed for related state estimation problems in similar appliations. Souy[16] used an extended Kalman filter to estimate the ure state in auto-lave uring of thik setion omposite laminates. The estimator onstrution assumed the online measurement of the distributed temperature state. For a similar problem, Parthasarathy, et al[14] introdued a nonlinear observer to estimate primarily the distributed temperature state. The ure state is subsequently omputed using the open-loop ure kinetis model from the estimated temperature. In both of the above works, the estimator design was based on an approximated ODE version of the oupled PDE-ODE proess model. Despite the ODE approximation error, the diret estimator of ure state in [16] is potentially more robust than the approah in [14] in the presene of disturbanes and errors in guessing the initial state. However, the approah in [16] is useful only for a part thikness(depth) large enough to aommodate plaement of a large number of temperature sensors aross the thikness. This is impratial for UV uring, as there is a fundamental limitation on the thikness of parts that an be UV-ured due to the above-mentioned UV attenuation with depth. A desirable observer for the UV uring appliation is an augmented or full-order observer of the oupled distributed temperature and ure state with the available boundary temperature measurement. However, the problem posed as suh laks observability as will be disussed in detail below. As a work around, in this paper, we design a two step temperature and ure state observer. First, the infinite dimensional temperature observer is designed based on boundary PDE bakstepping tehniques[17] to generate a bounded estimate of the distributed temperature state. Then, this estimated temperature information is used to onstrut a nonlinear ure state observer with some guaranteed onvergene onditions established via a Lyapunov analysis. The remainder of the paper is organized as follows. Setion II presents the problem statement onsidering a proess model for UV uring a omposite laminate. Setion III disusses the observability of the problem and details the two-step observer designs. Setion IV provides demonstrative simulation results. Setion V gives the onlusions of the paper. NOMENCLATURE : Speifi heat of omposite [ J g C ] p E : Ativation energy [ J mol ] 2 h : Convetive heat transfer [ W m C ] 2 I : UV-light intensity at surfae [ W m ] k : Thermal ondutivity of omposite [ W m C ] z L : Cure level observer gain [-] l : Sample thikness [ m ] m& n : Reation orders [-] p x& p : Temperature observer gains [-] 1 1 p& q : Constants parameters [-] R : Gas onstant [ J mol K ] s : Photoinitiator onentration [ wt % ] T z, t : Temperature distribution in physial spae [ C ] T abs : Absolute temperature [ K ] T : Ambient temperature [ C ] t : Time [ se ] w x, t : Temperature in normalized spae [ C ] wˆ x, t : Temperature estimate in normalized spae [ C ] ( ) : Temperature estimation error [ C ] w x,t x : Spatial diretion for normalized spae [-] z : Spatial diretion for physial spae [ m ] zt, : Cure level distribution in physial spae [%] xt : Cure level for estimated temperature [%] wˆ, ( ) : Cure level estimation error [%] a x,t H r : Polymerization enthalpy of resin [ J g ] : Absorptivity onstant [-] : Absorptivity of photoinitiator wt % m 1 v r : Volumetri fration of resin [-] 3 : Density of omposite [ g m ] 2 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

3 3 r : Density of resin [ g m ] 1 : Pre-exponential fator of rate onstant [ se ] II. PROBLEM STATEMENT We onsider a 1D proess for UV uring a fiberglass omposite laminate. A shemati of the proess set up is shown in Figure 1. The basis for the model is that of a UV uring proess model verified for uring a pure unsaturated polyester resin for a stereolithography/photo-fabriation appliation [6]. The following three onsiderations are added to the basi model. First, a resin volume fration fator is introdued in the proess model to onsider the fat that only the resin portion undergoes the photopolymerization reation([7],[14]). Seond, we take the average thermal properties of the resin and of the fiber for the properties of the omposite laminate in the z- diretion, whih is the indiated diretion of sample depth/thikness. Third, we model the attenuation of UVradiation in the resin-fiber matrix in the z-diretion aording to Beer Lambert s Law, where a single attenuation onstant will be taken for the laminate. This essentially assumes a uniformly wetted fiberglass and resin where the refrative indies of the fiber and resin are mathed. To ontinue with this assumption for general ases, the attenuation onstant for the ombined resin-fiber matrix may need to be modified[18]. UV Radiation where and p are the density and speifi heat apaity of the omposite laminate, respetively; k z is the thermal T z t is ondutivity of laminate in the z-diretion;, temperature distribution at depth z and time t ; vr is volumetri fration of resin in the omposite matrix; r is density of resin; and Hr is polymerization enthalpy of resin onversion; E is ativation energy, s is photoinitiator onentration is preexponential fator of rate onstant; R is gas onstant; I is UV- T z, t is absolute temperature in Kelvin; light intensity; abs zt, is ure level/state distribution; m& nare reation orders; p& q are onstant exponents; is the absorption oeffiient of photoinitiator; is absorptivity onstant of the UV-radiation at the boundary; h is onvetive heat transfer at the top boundary; l is the thikness of omposite sample, and d z, t dt is the rate T is onstant ambient temperature; and of ure onversion (rate of polymerization). The various domains and boundaries are: T, lx,,,l, 1 x,, and 2 l x,. We simplify the proess model (1) by introduing the following hange of variables: z x (2) l,, (3) w x t T x t T Fig. 1. Shemati of a UV Curing Proess. Considering the above onsiderations, the uring proess model is given by the following oupled PDE-ODE systems along with the boundary onditions and initial onditions: 2,,, T z t T z t d z t p kz v 2 rh rr int t z dt d z, t m n K z, T z, t1 z, t int dt q p K z, T s I exp E RTabs z, texpspz int T(, t) kz * I( t) h( T (, t) T ) in1 (1) z T ( l, t) in2 z T z, T ( z) in z, ( z) in The simplified model is summarized in the following form:,,,,, x, b, b x 1,,,,,,, w x t w x t f x t w in w t w t I in1 w t in2 w x w x in x t f x t w int x x in t xx a T The nonlinear term f represents the ure onversion rate and is given by: p E f x, t, w, ai exp * Rwabs x, t n x, t1 x, t exp ax m where 2 q k l, v H, psl, s, hl k, b lk (4) (5) p a r r p a a b wabs x, t w( x, t) T 273.The notation, w, w,, and w represent d dt, w t, w xand 2 w x 2,respetively. xx Note that even though the above model has been disussed for a omposite laminate, the essential proess onsiderations are quite similar for other UV uring appliations. One an t x 3 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

4 diretly use this model (by modifying the added onsiderations) and the disussions that follow for other UV uring appliations in advaned manufaturing, suh as photo-fabriation, 3D printing, uring oatings, et. III. PROPOSED OBSERVER STRUCTURE AND DESIGN The primary objetive of the observer is to estimate the zt, using the available boundary distribution of ure state temperature measurement. On initial onsideration, this objetive might seem easily ahievable by simply onstruting a full state observer for both the temperature and ure level states. However, as will be shown in subsetion A below, the full state is not observable using the adopted model with only boundary measurement of temperature. To overome this diffiulty, we propose the two-step observer design shown shematially in Figure 2. First, a temperature observer is designed to generate a bounded estimate of the temperature distribution aross the laminate. This estimated temperature distribution is then used in the ure state observer. CURE LEVEL OBSERVER TEMPERATURE OBSERVER SURFACE TEMP Figure2. Observer Design Struture UV UV-Composite Curing Proess A. Observability Cheks For a linear system, observability is a global property for whih there are well known riteria to hek. For nonlinear systems suh as the present appliation, the notion of observability depends on the regions of the state spae where the system operates and as suh it remains a loal property. Several versions of loal observability definitions and heks are available of whih we apply the ones by Zeitz[19], also summarized in[2], to the present uring proess model. To hek the observability of the full PDE-ODE system model for the uring proess, we first derive an augmented nonlinear ODE system by applying entral differene approximations for the spatial derivative in the PDE of (4). The resulting system dynamis and measurement equations an be written in the form: t h ( t), u( t), 1 t y t 1 w T where 1 n X is the augmented state vetor of the temperature and ure state at eah spatial disretization loation, and u I p U is a salar UV input, is the initial state vetor and y is the boundary temperature measurement. and are onneted open sets. h is a vetor funtion of the spatially disretized state and input, and n is dimension of the augmented state, whih is twie the size of the number of spatial disretization nodes adopted. An ODE system of form (6) is said to be loally observable [2], if y dy dt rank n 1 d y n 1 dt * o, u n n1 b, u,, u,where u, b 1,, n 1 is the th point. b time derivative of u, and * o, u (6) (7) is nominal operating Applying this test to the uring proess model with a threenode disretization, (and, so a 6 th order system: three for temperature and three for ure state), one finds that the observability matrix in (7) is of rank 3. It an be shown analytially that, for any disretization level, the rank is half the size of the full state spae. So, the full state is not observable. The two-part observer design is proposed to alleviate this diffiulty. B. Temperature Observer Design With only boundary temperature sensing pratially available for the UV uring proess, we find that the bakstepping boundary PDE observer design methods detailed in[17], among many others reviewed in [21], to be suitable for onstruting the temperature observer. Sine the bakstepping design methods are developed primarily for linear PDEs, we exploit the observation that the present temperature PDE is a semi-linear PDE that is lose to a linear one. Furthermore, sine this PDE is oupled to the ure state dynamis, in order to proeed with the temperature observers design, we make the pratial assumption that the rate of ure onversion d x, t dt is upper-bounded. Using the bakstepping approah, the temperature observer for proess model (4) with surfae (boundary) temperature sensing at the top w, t is onstruted as follows: 4 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

5 ,,,, wˆ ˆ ˆ t x t wxx x t p1 x w t w t ˆ ˆ a f x, t, w, w ˆ int wˆ,, 1, ˆ x t bw t p w t w, t bi on1 wˆ x 1, t on2 w ˆ x, wˆ ( x) in ˆ, ˆ,, ˆ w x t f x t w, wˆ int w ˆ x, w ˆ( x) in where, wˆ x, t is the estimated state of, the observer gains, wˆ, w x t, (8) p x & p are 1 is the ure state omputed using the ure state ODE as an open-loop observer whih uses the estimated temperature. The nonlinear term ˆf represents the ure onversion rate in terms of estimated temperature and is given by: ˆ p E f x, t, wˆ, w ˆ ai exp * Rwˆ abs x, t n x, t1 ˆx, t exp x m wˆ w a Defining the temperature observer error variable as w x, t w x, t wˆ x, t, and the ure state error x, t x, t x, t, and subtrating (8) from (4), the wˆ wˆ observer error dynamis beomes:,, 1,,,, ˆ,, ˆ x,, x 1,, ˆ,,,, ˆ,, ˆ, ( ) w x t w x t p x w t f x t w w in w t p1w t on1 w t on2 w x w x in w x t f x t w w w int wˆ x wˆ x in t xx w T where, f x, t, w, wˆ,, f x, t, w, fˆ x, t, wˆ, wˆ a wˆ (9) (1). Note that, the distributed temperature error dynamis in (1) onsists of two parts: the first two terms onstitute the linear part and the last term f (ure onversion rate error) onstitutes the nonlinear part. Fat 1: Treating f as unknown bounded disturbane, the observer designed to exponentially stabilize the linear part of the error dynamis PDE in (1) (ignoring f ) leads to bounded temperature estimation error when applied to the full nonlinear PDE error dynamis (1) (inluding f ). A proof is provided in Appendix A. The design of the observer for the linear part of the PDE an be done following the approah detailed in [17]. The linear part of the of observer error dynamis (1) is given by: ( x,t) = xw xx ( x,t) - p 1 ( x)w(,t) inw T ( ) = - p 1 w(,t) ong 1 ( ) = ong 2 ( ) = w ( x) inw ìw ï t ïw x,t í ïw x 1,t ï îï w x, The observer gains 1 1 (11) p x & p an be determined by transforming the error system (11) to a stable target system (13) using the state transformation: w x t vx, t,, p x y v y t dy (12) The hoie of partiular stable target system is not unique but the following exponentially stable target system is onsidered onvenient with tuning parameter :,,, x, x 1,, vt x t vxx x t v x t int v t on1 v t on2 v x v x in (13) The proof for the exponential stability of a similar target system as (13) is provided in our previous work[22] and also in [23]. We use the transformation in (12) along with (11), to derive p x, y the following onditions for the observer gain kernel and observer gains: pxx x y pyy x y p x y,,, (14) p 1, y (15) x 2, x 1 (16) p x x The observer gains are: (17) p1 x py x, p (18) p1, Solving (14-16) analytially, the observer gain kernel and ultimately the gains an be shown to be:, 1 y p x y I1 2 x yx y 2 x yx y (19) I1 x2x I2 x2 x p1 x x2 x x 2 x (2) p1 2 (21) 5 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

6 Using these gains in the temperature observer PDE (8) along with the open-loop ure state estimates, we an generate the distributed temperature estimate, with bounded estimation errors via Fat 1. C. Cure State Observer Design For brevity of notations, we drop the independent variables t and x in the ure state dynamis and we rewrite it as follows: a = f w,a ( ) (22) Before we proeed, note that (22) is to be spatially disretized along the same disretization as adopted for implementing the temperature observer PDE (8). Therefore, with some abuse of notation, a is the ure state vetor indexed by spatial loation x[,1 p,,1] T where p is the number of disretization points, and therefore, the size of the ure state vetor. The funtion f w,, whih represents the ure onversion rate, an be fatored into nonlinear funtions of input and state as follows:, rw g (23) f w where p m r w ai expe Rwabs exp a x, and g 1 n. For strongly nonlinear systems like those of the ure dynamis (22), a ommon estimation approah is the extended Kalman filter (EKF) [24, 25]. However, the omputational ost of reursively omputing the gains is high and verifying the suffiient onditions for stability is not trivial. To overome these defiienies, we propose a variable struture ure state observer design where Lyapunov analysis is used to establish theoretial onvergene onditions and then proess speifi modifiations are applied for implementation. Assuming that the estimated temperature from the temperature observer is suffiiently aurate (by the observer design), we proeed with the ure state observer design by treating ŵ» w. The ure state observer is then onstruted using the proess dynamis (22) and supposing that a pseudomeasurement signal y = a an be obtained by inverting the temperature estimate obtained via (8). Under these onsiderations, the pseudo-measurement signal an also be written as: y f wˆ, (24) Given (22) and (24), we an onstrut the observer as follows: ˆ, L y ˆ f w ˆ ˆ (25) where the ure state estimate is denoted by ˆ and the observer gain L is a diagonal p p matrix. We introdue the following hange of variables[2] to later avoid the need for omputing the time derivative w in the pseudo-measurement (inversion of (8)), ˆ Z L wˆ (26) a Substituting (26) into (25), we reonstrut the observer (24) in the following implementable form: L L Z f wˆ, Z L ˆ 1 ˆ 1 ˆ a w L wxx p w w (27) a For stability analysis we re-write the observer dynamis in the form: ˆ, ˆ, ˆ, ˆ f w ˆ L f w f w ˆ (28) Introduing the estimation error variablea = a -â, subtrating (28) from (22), and substituting (23) the error dynamis takes the form: 1 L g, ˆ rwˆ (29) where g( a,â) = g( a) - g( â). Sine the elements in the ure state vetor are deoupled from eah other, we an design the diagonal observer gain elements separately by using an element-wise Lyapunov funtion andidate V i = 1 2 a i2, i = 1,..., p (3) Taking the time derivative of Lyapunov funtion (3) and substituting the element-wise error dynamis (29), 1, ˆ ˆ (31) V L g r w i i i i i where, rwˆ, wˆ. It is lear that to make V i < (stable observer), the gains L i have to swith sign based on the sign of g i ( a,â )a i. The latter is not readily pre-determined. We apply the following proess speifi onsiderations to overome this limitation of the observer design. Remark: I. Figure (3) shows the evolution of g as a funtion of the ure state. The maximum of g ours at m m n, whih is expliitly dependent only on the reation order onstants m and n. The funtion g is stritly inreasing funtion of until g reahes its peak value. Then after, it beomes a dereasing funtion of. Sine â is omputed/available, it an be determined that g ˆ lies in the first zone if â m é ëm + nù û and it lies in the seond zone ifâ > m é ëm + nù û. An estimate of g an be omputed from the pseudo-measurement (24) (i.e. g a dg dt ) a ( ) = y r( w) assuming ŵ» w). Then, the gradient ( an be omputed, and ombined with the observation that the ure rate y ³,"t, one an determine that g lies in the first zone if dg dt, and the seond zone if dg dt. 6 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

7 To hoose the observer gains, we onsider two ases: Case 1: If g and g ˆ are in the same zone: ( ) ³ in the first zone, and a i g i ( a,â) in the seond a i g i a,â zone. Therefore, to makev i, in the first zone we hose Li 1, and in the seond zone we hose Li 1. Case 2: If g and g ˆ are in different zones: the sign ( ) annot be predetermined; still the sign of Li an be ( a,â ) based on the knowledge of the a i g i a,â hosen as a sign of a i g i magnitudes of g and g ˆ. g ().3.2 First Zone Figure 3: Evolution of Funtion g with Cure State Remark II. In pratie, the proess starts from unured state. One an hoose initial ure state estimates near zero, and it is possible to have onvergene of the estimate using a onstant gain in the first zone. For the seond zone, the observer gain an be swithed to zero where the observer beomes open-loop stable. However, the latter removes a means of tuning the observer. Remark III. In the implementable form of the observer given in (27) there is an L term. While swithing the observer gain, this term may produe unaeptable spikes. To avoid this, one an simply plae saturation on this term (or replae the sign funtion with some smooth approximation). IV. RESULTS AND DISCUSSIONS In this setion, we present simulation results to demonstrate the performane of the proposed two-step observer. The simulation onsiders the 1D UV uring model (4) as the proess model. The assoiated thermal, hemial and material onstants for photopolymerization of unsaturated polyester resin are extrated from published work[6, 26]. For the fiberglass, E- glass thermal properties suh as thermal ondutivity, ( k.12 W m. o C ), speifi heat ( p.8 J g. o C ), and 3 Seond zone density ( 2.55 g m ) are used. Volume frations of 4% and 6% are used for fiber and resin, respetively to determine the average thermal properties of the omposite laminate. A thikness of 5 mm is onsidered for the omposite laminate. A 2 onstant UV-intensity of 6 mw m is used for the entire uring duration, whih is as long as 5 seonds. For solving the PDEs, a forward in time and entral in spae (FTCS) finite differene method was used for both the proess model and the temperature observer. A oarse grid of 11 nodes and a fine grid of 21 nodes were onsidered for the observer and proess models, respetively. In both respetive models, the same level of disretization was used for both the ure state and temperature state. The deay rate of the temperature observer error an be tuned with the free parameters imbedded in the observer gains of (2) and (21). For the ure state observer, the pratial ase in remark II is onsidered with a onstant gain element greater than one for all ure state elements in the first zone and then swithed to zero gain (open-loop observer) in the seond zone. The performane of the observer is illustrated in two ways. Figure 4 shows the spatial distribution and the evolution of the spatial 2-norm of the observation error for temperature state plotted on a faster time sale. The observation error for ure state is plotted similarly in Fig Time, se Fig.4: Convergene of Observer Error for Temperature State: (a) Spatial Distribution, and (b) Spatial 2-norm The results in Figs. 4 and 5 show that the initial temperature and ure state errors quikly stabilize in about 2 and 3 seonds, respetively. It should be realled that sine the ure state observer uses the output injetion term from the temperature observer, the performane of the ure state observer is dependent on the performane of the temperature observer. This an be verified by lose examination of Figs.4a &5a. For the bakstepping temperature observer with observer gain ( p 1 ) at top surfae boundary in addition to in domain gain ( p1 ( x ) ), the temperature errors near the top boundary onverge faster than those near the bottom boundary. The ure level observer error also follows the same trend sine it uses the estimated temperature. The performane of proposed two-step observer is also tested in the presene of measurement noise by adding a white, zero mean Gaussian noise on the boundary measurement. For a andidate boundary temperature sensor (IR-amera) with preision of 2 o C, we assumed the Gaussian noise to have a ovariane of this magnitude. Figures 6 & 7 show results for k ~w(x)kl Temperature Observer error Time, se Fig.5: Convergene of Observation Error for Cure State: (a) Spatial Distribution, and (b) Spatial 2-norm k ~,(x)kl Cure level Observer error 7 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

8 representative nodal points: top ( z ), middle ( z.5l ) and bottom ( z l ), respetively. In this ase, we onsidered a redued ure level observer gain to show the error onvergene after signifiant ure evolution is ahieved. The error onvergene is ahieved in the first zone as it is stated in remark II. T, o C T, o C T, o C T(z=)measurement T(z=)estimate 3 2 T(z=)atual T(z=.5l)estimate T(z=.5l)atual T(z=l)estimate T(z=l)atual Time,se Figure 6: Evolution of Atual and Estimate Temperature at the Top, Middle and Bottom Nodes along the Depth of the Laminate.,%,[%],% (z=) estimate (z=) atual (z=.5l) estimate (z=.5l) atual (z=l) estimate (z=l) atual Time,se Figure 7: Evolution of Atual and Estimate Cure level at the Top, Middle and Bottom Nodes along the Depth of the Laminate The results in Fig. 6 show that the estimate of temperature onverges to the atual state in a short time. However, the influene of noise in the measurement is more pronouned as we go from top to bottom. This is as a result of observer gain (19) inreasing in magnitude in the same diretion. This variation of observer gain along depth is shown in Fig. 8. Also, the result in Fig.7 shows the onvergene of ure state. It an be seen that the noise in the temperature measurement is not transmitted to the ure-state estimate. This is beause of the negligible effet of noise in the temperature as it enters the ure dynamis in the exponential form exp E Rwˆ abs. In the ure state observer, the noise in Z and L a wˆ anel eah other while omputing the ure level estimate via (26). p1(x) Observer gain, p1(x) Depth,[-] Figure 8: Variation of Observer Gain p1 x with Depth V. CONCLUSIONS This paper proposed an online estimation sheme for obtaining ure state information in radiative UV uring proesses. Considering the hallenge of observability from pratially available boundary temperature measurement, the paper outlined a two-step observer as a asade of a distributed temperature observer and a nonlinear ure state observer. Assuming bounds on the nonlinear ontribution of ure onversion on the temperature dynamis, bakstepping PDE observer design tehniques are applied to derive a distributed temperature observer. For ure state estimation, a variable struture nonlinear observer is designed assuming an aurate temperature estimate from the previous step. The performane of the proposed observer was tested through simulations of a proess model for UV uring of a fiberglass omposite laminate. The results showed that the proposed two-step observer performs well even in the presene of measurement noise. ACKNOWLEDGMENTS This researh has been supported, in part, by the National Siene Foundation under grant no. CMMI and the US Department of Energy s GATE Program. REFERENCES [1] I. F. P. Roose, C. Vandermiers, M.-G. Olivier & M. Poelman, "Radiation uring tehnology: An attrative tehnology for metal oating," Progress in Organi Coatings vol. 64, p. 8, 29. [2] M. F. Perry and G. W. Young, "A mathematial model for photopolymerization from a stationary laser light 8 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

9 soure," Maromoleular theory and simulations, vol. 14, pp , [3] P. Compston, J. Shiemer, and A. Cvetanovska, "Mehanial properties and styrene emission levels of a UV-ured glass-fibre/vinylester omposite," Composite Strutures, vol. 86, pp , [4] R. Shwalm, UV Coatings: Basis, Reent Developments and New Appliations: Elsevier, 26. [5] J.-Y. C. a. S.V.Hoa, "Fiber-Opti and Ultrasoni measurements for In-situ Cure Monitoring of Graphite/Epoxy Composite," Journal of Composite materials, vol. 33, p. 22, [6] J. M. Matias, P. J. Bartolo, and A. V. Pontes, "Modeling and simulation of photofabriation proesses using unsaturated polyester resins," Journal of Applied Polymer Siene, vol. 114, pp , [7] X. Wang, "Modeling of in-situ laser uring of thermosetmatrix omposites in filament winding," PhD Dissertation, university of Nebraska, 21. [8] F. E. Thau, "Observing the State of Nonlinear Dynamial Systems," International journal of ontrol vol. 17, [9] S. Raghavan and J. K. Hedrik, "Observer design for a lass of nonlinear systems," International Journal of Control, vol. 59, pp , [1] A. J. Krener and W. Respondek, "Nonlinear observers with linearizable error dynamis," SIAM Journal on Control and Optimization, vol. 23, pp , [11] S. V. Drakunov, "An adaptive quasioptimal filter with disontinuous parameters," Avtomatika i Telemekhanika, pp , [12] H. W. Sorenson, Kalman filtering: theory and appliation vol. 38: IEEE press New York, [13] S. J. Julier and J. K. Uhlmann, "A new extension of the Kalman filter to nonlinear systems," 1997, pp [14] S. Parthasarathy, S. C. Mantell, and K. A. Stelson, "Estimation, ontrol and optimization of uring in thiksetioned omposite parts," Journal of dynami systems, measurement, and ontrol, vol. 126, pp , [15] J. Baker and P. D. Christofides, "Finite-dimensional approximation and ontrol of non-linear paraboli PDE APPENDIX A systems," International Journal of Control, vol. 73, pp , 2 2. [16] K. A. Souy, "Proess modeling, state estimation, and feedbak ontrol of polymer omposite proessing," Washington Univ., Seattle, WA (United States) [17] M. Krsti and A. Smyshlyaev, Boundary ontrol of PDEs: A ourse on bakstepping designs vol. 16: Soiety for Industrial Mathematis, 28. [18] A. Endruweit, W. Ruijter, M. S. Johnson, and A. C. Long, "Transmission of ultraviolet light through reinforement fabris and its effet on ultraviolet uring of omposite laminates," Polymer Composites, vol. 29, pp , [19] M. Zeitz, "The Extended Luenberger Observer for Nonlinear Systems," Systems & Control Letters, vol. 149, [2] M. Soroush, "Nonlinear state-observer design with appliation to reators," Chemial Engineering Siene, vol. 52, pp , [21] Z. Hidayat, R. Babuska, B. De Shutter, and A. Nunez, "Observers for linear distributed-parameter systems: A survey," 211, pp [22] A. Yebi and B. Ayalew, "Feedbak Compensation of the In-Domain Attenuation of Inputs in Diffusion Proesses," in Amerian Control Conferene (ACC), Washington, DC, USA, 213, pp [23] A. Smyshlyaev and M. Krsti, Adaptive ontrol of paraboli PDEs: Prineton University Press, 21. [24] K. Reif, S. Günther, E. Yaz Sr, and R. Unbehauen, "Stohasti stability of the disrete-time extended Kalman filter," IEEE Transations on Automati Control, vol. 44, pp , [25] K. Reif, F. Sonnemann, and R. Unbehauen, "An EKFbased nonlinear observer with a presribed degree of stability," Automatia, vol. 34, pp , [26] P. Bártolo, "Optial approahes to marosopi and mirosopi engineering," University of Reading, 21. PROOF FOR TEMPERATURE OBSERVER In this setion, a brief proof is provided for Fat 1. Starting V t with a Lyapunov funtion andidate: ξp w t ξ w x t dx p x w t w x t dx 1 V t 2 w x t dx A1 + f x t w w α α w x t dx A2 Taking the derivative of (A1), applying PDE &BCs in (1), and integrating by parts: Assuming an upper bound on f x, t, w, wˆ,, wˆ F x, we have 9 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use:

10 V t ξp w t ξ w x t dx p x w t w x t dx + F x w x t dx A3 Equation (A3) an be re-written by substituting (2) and (21) in A(3) V t 2 w t ξ w x t dx N x w t w x t dx + F x w x t dx A4 where I1 x2x I2 x2 x N x, for x,1. x2 x x 2 x For the seleted exponential stable target system (12), the linear part in (A4) an be made negative with proper seletion of design parameters leading to the exponential onvergene of the linear part of the PDE(9) [17]. However, due to the nonlinear part in (A4), the estimation error may not onverge to zero. However, the Lyapunov derivative V will be negative until the following ondition is satisfied: 2 w t ξ w x t dx N x w t w x t dx > F x w x t dx This essentially means that the estimation error will reah to a ertain bounded manifold and will subsequently stay there. The manifold size is determined by the seletion of design parameters and size of the bound on the nonlinear term F(x) With seletion of high values of, the steady-state estimation error an be made suffiiently small. 1 Copyright 214 by ASME Downloaded From: on 5/31/215 Terms of Use: