Control of a ceramic tiles cooling process based on water spraying

Journal of Process Control 19 (2009) 1073–1081

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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

Control of a ceramic tiles cooling process based on water spraying R. Sanchis *, I. Peñarrocha Departament d’Enginyeria de Sistemes Industrials i Disseny, Universitat Jaume I, Campus de Riu Sec, 12071 Castelló, Spain

a r t i c l e

i n f o

Article history: Received 25 January 2008 Received in revised form 23 January 2009 Accepted 24 January 2009

Keywords: Tile temperature control Water spraying Industrial adaptive control Feedforward control Event-driven measurements

a b s t r a c t The control of the surface temperature of ceramic tiles in a real industrial production line is developed. The process consists of a transportation band that carries the hot tiles through a water sprayer whose objective is to reduce its temperature. Two input signals can be modified: the velocity of the transportation band and the flow rate of the sprayer. In order to control the outlet surface temperature, the quantity of water deposited (and hence evaporated) per tile, that is a static function of the velocity and the flow rate, is used as the control input. The inlet temperature has a tile to tile fast pattern variation and a slow average change. First, the experimental identification of the process model is carried out. Then a feedback PI controller based on the measurement of the outlet temperature is then tested, showing a good average tracking, but a poor compensation of the fast variations. An adaptive feedforward control based on the measurement of the inlet and outlet temperatures is developed and also tested in the plant, showing a much better performance, but a higher cost. Finally, a disturbance observer based feedforward control is tested, showing an intermediate performance and cost. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction On the ceramic tiles production lines there is commonly a water sprayer after the dryer and just before the glazer. The objective of this sprayer is to reduce the temperature of the tiles (by evaporation) to adequate levels for the glazer to work well. An inadequate temperature may produce defects that lead to the rejection of the produced tile. Too high temperatures produce a kind of surface defects called ‘‘punctures”, while too low temperatures produce surface defects called ‘‘pools”. Nowadays, almost all the ceramic plants have a fixed water sprayer, that can only be changed manually by the operator. The only automatic regulation system that is used on some plants consists of an extra sprayer that can be switched on or off by an electrovalve depending on the inlet temperature. The measurements obtained in a real plant under normal production show that the temperatures are highly time varying. As a consequence, a fixed sprayer (or a discrete sprayer switch) leads to very important fluctuations on the temperature of the tiles that enter the glazer. This paper is concerned with the study of the automatic control of the surface temperature by continuously changing the mass of water deposited (and hence evaporated) per tile. The objective is to maintain the inlet temperature at the glazer as uniform as

* Corresponding author. Tel.: +34 964728175; fax: +34 964728170. E-mail addresses: [email protected] (R. Sanchis), [email protected] (I. Peñarrocha). 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.01.007

possible in order to reduce the percentage of defects associated to temperature variations. There are few works in the literature about the water spraying evaporative cooling. There is one directly related to ceramics [1], where the transient cooling of ceramic solids by water drops evaporation is studied theoretically. Another application that is interesting due to its similarity with the proposed problem is the cooling by water spraying in steel continuous casting process [2]. Nevertheless, the conditions and the temperature ranges are very different from the studied case. There are also other theoretical studies about the evaporative cooling models [3,4], but the complexity of the process and the particularities of the case studied makes the direct experimental identification more suitable. There are also works in the literature dealing with several aspects of the ceramic tiles production, as the automatic inspection of tiles by artificial vision [5], or control of roller kilns [6], or the development of sensors to measure different variables [7], or about control technologies for the tile industry [8], but none of them deal with a control problem similar to the one treated in this paper. The paper deals with the solution of the industrial control problem based on standard well known techniques for identification [9,10], and for the control algorithms [11,12]. The layout of the paper is as follows: in Section 2, the process to be controlled and the experimental setup (sensing and actuating procedures) are described. In Section 3, the process identification is carried out. In Section 4, a feedback strategy is stated and developed, and in Section 5, the adaptive control is developed and tested improving the results. In Section 6, a disturbance observer

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feedforward control is defined and simulated. Finally in Section 7 the conclusions are summarized. 2. Process description and experimental setup The industrial process is shown in Fig. 1, including the implemented control system. It consists of a tiles transportation band with a clean water sprayer that creates a flat water curtain perpendicular to the band. When one tile passes through the water curtain a given quantity of water is deposited on its surface. Due to the high temperature of the tile, the water evaporates extracting a given quantity of heat and hence reducing the surface temperature. The temperature reduction depends on the quantity of water deposited per tile, that depends on the velocity of the tile and on the flow rate of the water nozzle. The nozzle is situated inside a steel cabinet. The inlet and outlet temperature sensors are located outside this cabinet, and hence they are separated from the nozzle position (around 2 m away). The water sprayer is situated on the ceramic tile production line between the dryer and the glazer processes. The dryer works in such a way that the outlet temperatures of the tiles (that are the inlet temperatures for the process to be controlled) have important variations with time. On one hand there are important differences from one tile to the next, due to their different position inside the dryer, and on the other hand there are slower changes on the average temperature along time. In Fig. 2 the temperatures before and after the water sprayer in a normal operation are shown when the quantity of water per tile is fixed (i.e. no control implemented). It is clear that the fast temperature variation between tiles follows a ”periodic” pattern, that is related to the position of the tiles inside the dryer, as it is shown in Fig. 3. The period and shape of this pat-

tern depends on the tile size, because the tiles are grouped in planes inside the dryer. Fig. 4 shows the magnitude of the frequency spectrum of the inlet temperature signal during the identification experiment. High values can be observed at low frequencies (slow temperature variation), and two different peaks at frequencies 0.25 and 0.50 samples1, corresponding with the repetitive temperature pattern of 2 and 4 tiles observed in Fig. 3. The outlet temperature has a very important variation following the pattern of the inlet temperature. The objective of the control system studied in this paper is to minimize the variation of that outlet temperature, i.e. to maintain it as uniform as possible, by means of applying a different mass of water on each tile. The experimental setup of the implemented control system is shown in Fig. 1. In order to measure the inlet and outlet temperatures two infrared non contact sensors are used, one before the nozzle and the other one after it (with a sufficient separation that guarantees the evaporation of all the water before reaching the outlet sensor). Two optical proximity sensors detect the presence of the tiles and allow the synchronization of the temperature measurements. The equivalent temperature of a tile is calculated as the average of several measurements taken after the settling time of the sensor. This settling time is around 200 ms, hence the tile should be at least 0.4vmax m long (where v max is the maximum velocity that will be applied) in order to be able to take valid measurements over half the length of the tile. The velocity of the band can be changed by means of a variable speed drive, while the water flow rate can be varied by a high pressure piston pump driven by a variable speed drive and AC motor. The volumetric nature of the pump implies that the flow rate is proportional to its rotation speed. The control algorithm is implemented on an industrial computer with a data acquisition card that receives the analog and digital signals and produces the analog outputs for the variable speed drives. The settling time for the change in velocity is about 200 ms,

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but the settling time for the change in water flow rate is approximately 450 ms. This settling time limits the velocity and the minimum tile separation, because the velocity and the flow rate must be in steady state before the tile starts passing under the sprayer. In order to reduce the settling time for the flow rate, an open loop feedforward lead compensation of the flow rate signal is implemented. It consists of a discrete lead filter (implemented at a fixed sampling time of 1 ms) that takes as input the desired flow rate, and whose output is the flow rate signal sent to the variable speed drive of the pump. The transfer function from the signal sent to the variable speed drive to the real flow rate has been identified experimentally leading to the discrete transfer function (with a sampling period of T ¼ 1ms

GðqÞ ¼

9:0333  107 ð1 þ 3:681q1 Þð1 þ 0:2643q1 Þq1 ð1  0:9858q1 Þð1  0:9847q1 Þð1  0:9753q1 Þ

ð1Þ

The feedforward compensator term has been calculated cancelling the poles of the model and fixing one discrete pole on 0.98 (equivalent to a continuous pole with a 200 ms settling time) and with a unitary gain, leading to

CðqÞ ¼ 3727

ð1  0:9858q1 Þð1  0:9847q1 Þð1  0:9753q1 Þ 1  0:98q1

ð2Þ

This compensation term does not affect the steady state value of the flow rate applied to the tile, but reduces the settling time of the change in flow rate to approximately 200 ms, and hence, the minimum distance between tiles in order to apply a uniform water layer over the whole tile surface, is 0:2 v max m (instead of 0:45v max m, as would be without the compensation term). In order to apply the correct mass of water to a given tile it is necessary to know the exact instant when it passes under the nozzle. This is accomplished simply by integrating the velocity of the tile with time (the exact distance between the proximity sensor and the nozzle is known). This idea is also used to relate the outlet temperature measurements to inlet temperatures and mass of water applied to a given tile. In order to develop the tile temperature control system, the process variables are defined as: - k. Tile number, discrete ‘‘time” index used for the signals, but not related to time, but to the number of tiles passing through the process. - T in ðkÞ. Average inlet surface temperature of tile number k. - T out ðkÞ. Average outlet surface temperature of tile number k. - Q ðkÞ. Flow rate of water while tile number k is passing under the nozzle. It is assumed that, after the change in flow rate from the previous tile ðQ ðk  1ÞÞ, the flow rate has reached the steady state value Q ðkÞ before the new tile starts passing under the nozzle. - v ðkÞ. Velocity of tile number k passing under the nozzle. Again, it is assumed that, after the change in velocity from the previous tile ðv ðk  1ÞÞ, the velocity has reached the steady state value v ðkÞ before the new tile starts passing under the nozzle. ðkÞ . Control action. This variable is proportional to the - uðkÞ ¼ Qv ðkÞ mass of water deposited on tile k. The mass deposited per tile, that depends on the water flow rate and the velocity of the tile, is the control input to be calculated by the controller. Once it is known, there are two redundant actuation signals to produce the desired control action: the velocity or the flow rate or both could be changed to lead to the same deposited mass. In order to define a strategy to solve this redundancy, the limitations on the two actuation signals must be taken into ac-

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count. The limitations on the different variables that must be applied are -

v ðkÞ. There is a minimum velocity limit due to the minimum tile separation needed (v min ). There is also a maximum velocity limit given by the band AC motor maximum velocity (v max ).

There is also a limitation in the velocity change from one tile to the next ðDmax Þ, because consecutive tiles may come close leading to failures and line stops in some processes down the line. - Q ðkÞ. There is a minimum flow rate under which the water nozzle does not produce a continuous curtain ðQ min Þ. There is also a maximum flow rate limited by the maximum pump speed ðQ max Þ. ðkÞ . There is a minimum mass of water given by the max- uðkÞ ¼ Qv ðkÞ min Þ. There is imum velocity and the minimum flow rate ðumin ¼ Qv max also a maximum mass of water that can be applied because of puddle formation due to water excess. This maximum mass of water is lower than the one that corresponds to maximum flow Þ. rate and minimum velocity ðumax < Qv max min Due to the important temperature changes between consecutive tiles, and the velocity change limitation, the velocity signal is not suitable for tile to tile control. As a consequence, the water flow rate will be used for this purpose. The velocity will be slowly changed in order to maintain the flow rate approximately centered in its range. Once the control algorithm determines the desired control action ðuðkÞÞ, the velocity and the water flow rate are obtained according to the following idea: if uðkÞ is saturated, or the calculated flow rate with the previous velocity ðQ ðkÞ ¼ uðkÞv ðk  1ÞÞ is out of the flow rate range, then change the velocity accordingly. In other case, leave the velocity unchanged. The detailed algorithm can be described as: – if ðumin 6 uðkÞ 6 umax Þ and ðQ min 6 uðkÞv ðk  1Þ 6 Q max Þ then  v ðkÞ ¼ v ðk  1Þ  Q ðkÞ ¼ uðkÞv ðkÞ – else if ðuðkÞ > umax Þ then   max  v ðkÞ ¼ max uuðkÞ v ðk  1Þ; v ðk  1Þ  Dmax ; v min  if ðumax v ðkÞ < Q max Þ then Q ðkÞ ¼ umax v ðkÞ  else Q ðkÞ ¼ Q max  max v ðkÞ ¼ max Qumax ; v ðk  1Þ  Dmax ; v min – else if ðuðkÞ < umin Þ then   min  v ðkÞ ¼ min uuðkÞ v ðk  1Þ; v ðk  1Þ þ Dmax ; v max  if ðumin v ðkÞ > Q min Þ then Q ðkÞ ¼ umin v ðkÞ  else Q ðkÞ ¼ Q min  min v ðkÞ ¼ min Qumin ; v ðk  1Þ þ Dmax ; v max – else if ðuðkÞv ðk 1Þ > Q max Þ then  max ; v ðk  1Þ  Dmax ; v min  v ðkÞ ¼ max QuðkÞ  Q ðkÞ ¼ Q max – else if ðuðkÞv ðk 1Þ < Q min Þ then  min ; v ðk  1Þ þ Dmax ; v max  v ðkÞ ¼ min QuðkÞ  Q ðkÞ ¼ Q min In this way, as can be observed in Fig. 12, if the control action or the water flow rate saturate, the velocity changes in order to center the flow rate in its range for the next tiles. This leads to the use of the maximum range in the regulation of the water mass per tile.

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2.1. Control strategies

In the next sections, these strategies are developed in detail.

3. Process identification The complexity of the thermal process makes unsuitable a theoretical identification, hence an experimental identification is proposed. For this purpose a different mass of water is applied to each tile, and the data of inlet and outlet temperatures together with the mass of water deposited is stored for each tile. In Fig. 5, the exciting  control input vq and the measured temperatures are shown. The objective of the identification experiment is to obtain a static function relating the inlet and outlet temperatures to the control action as

  QðkÞ : T out ðkÞ ¼ f ðT in ðkÞ; uðkÞÞ ¼ f T in ðkÞ; v ðkÞ

ð3Þ

First, a linear function was fitted by least squares leading to

T out ðkÞ ¼ 44:8 þ 0:548T in ðkÞ  16:3uðkÞ:

ð4Þ

In Fig. 6, the residues of the estimated function are plotted with respect to the sample (number of tile), and the independent variables on the model (the control input, and the inlet temperature). It is clearly shown that the residues with respect to the control input

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- Outlet temperature sensor and PI controller. This is the lowest cost system, but its performance is limited because it only regulates the slow ”average” temperature change (it cannot make tile to tile regulation). This solution can be valid when the type and size of tiles make the tile to tile difference not too large. - Inlet and outlet temperature sensors and adaptive controller. This is the highest cost system, but the highest performance. If only the inlet temperature sensor were used, a static feedforward control could calculate the amount of water to be applied to each tile as a function of its inlet temperature (using a static process model identified off line). However, the absence of feedback would produce a large error due to disturbances and process change with time. If the inlet and outlet temperature sensors are used, the process model can be identified on line and therefore, it can be updated continuously. Then, the feedforward control with the continuously updated model leads to a low tracking error. The outlet temperature sensor is only used to update the process model, but this introduces an implicit feedback that reduces the error. - Outlet temperature sensor and observer based feedforward controller. In order to approach the performance of the adaptive strategy, but with a cost near the PI alternative, an observer based feedforward strategy is developed, where only the outlet sensor is used. An observer uses the outlet temperature measurement to predict the future inlet temperatures, based on a disturbance model identified off line. Then, this predicted inlet temperature is used to calculate the necessary amount of water to each tile using a static process model. The performance is better than the PI because there is an approximate tile to tile regulation, but is worse than the adaptive strategy because the disturbance and process models are not updated.

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One of the most expensive elements in the control system are the infrared temperature sensors (pyrometers). In order to develop alternative control solutions with different cost, several configurations have been analyzed, using a different number of temperature sensors (inlet sensor, outlet sensor, or both). The idea is to compare the implementation cost and the achieved performance. The configurations that are analyzed are:

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Fig. 6. Residues of the identification experiment plotted against sample k, w.r.t. u ¼ Qv and against T in . Linear model (Eq. (4)).

are not white noise (the deterministic pattern is evident). In order to improve the model, an adequate nonlinear function must be found. The easiest approach [10], is to increase the degree of the polynomial. Adding a quadratic term of the control action and applying least squares one obtains

T out ðkÞ ¼ 58:8 þ 0:531T in ðkÞ  44uðkÞ þ 14:9uðkÞ2 :

ð5Þ

The residues are now much more independent, but the function has four parameters instead of three. The standard deviation of the error is 1.56 °C. In order to reach a similar error behavior but without increasing the number of parameters, several nonlinear transformations on the variable uðkÞ were tried. The result was the following non linear (but linearly parameterized) function with three parameters (also fitted by least squares).

T out ðkÞ ¼ a þ bT in ðkÞ þ c

1 1 ¼ 18:4 þ 0:531T in ðkÞ þ 11:5 ð6Þ uðkÞ uðkÞ

The low number of parameters was especially important for the implementation of a robust adaptive control (as will be shown in next sections). In Fig. 7, the residues of the estimated surface are plotted against the inlet temperature, the number of samples and the inverse of the control action. In the figure, the low correlation

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Fig. 7. Residues of the identification experiment plotted against sample k, w.r.t. and against T in . Nonlinear model (Eq. (6)).

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of the error with those variables is shown. The standard deviation is now 1.54 °C. This implies that there are important stochastic effects that cannot be compensated (i.e. a limitation on the achievable performance). Other non linear functions were tried leading to a slightly lower error variance, but the increase in complexity (and hence the loss of robustness) did not compensate the accuracy improvement. 4. Feedback control The first control configuration to be analyzed is the lowest cost. Only the outlet temperature sensor is used to define a feedback controller. The idea is to regulate the slow ‘‘average” value of the outlet temperature by means of a slow variation on the control action instead of applying a specific control action for each tile. This controller does not require tracking the position of the inlet tiles because the water is not applied to any specific tile, and, therefore, only one detector and one infrared pyrometer placed on the output are needed, leading to a low cost implementation. The water mass to be applied is calculated from the outlet temperature measurement and the control action is applied asynchronously with the pass of the tiles through the water curtain. For this reason, this implementation cannot regulate the temperature variations between consecutive tiles. Nevertheless, for some tile types and dryers, this variation between tiles may be small, and the approach could reach an acceptable performance. Updating the control input synchronously with the outlet temperature measurement (when a tile crosses completely the outlet pyrometer) introduces a transport delay from input to output, because the new control action effect is shown N tiles later on the outlet temperature sensor as there is a significant physical separation from water sprayer to outlet sensor (see Fig. 8). Index k now

applies to the number of measured outlet tiles, and the time variant delay N is the number of tiles that lie on the transportation band between the sprayer and the measurement point at the instant where the tile temperature is measured. N can vary between N ¼ 1 and a maximum value N max that depends on the tile size and the distance between the sprayer and the sensor. The time varying nature of N is due to the non uniform tile to tile separation. For the tiles where the experiment has been developed these delays can take a value between N ¼ 1 and N ¼ 4 tiles. As the only dynamics present on the system is a varying delay, a PI controller is proposed to robustly correct the slow variation of the mean temperature. Now a transfer function between control action and outlet temperature must be obtained using the possible delay and the static relationship between control action and outlet temperature exposed in (6). Let us call xðkÞ ¼ 1=uðkÞ the system manipulated input. Then the approximated linear difference equation that relates the control action and the outlet temperature for the experimented production line can be obtained linearizing (6), leading to

DT out ðkÞ ¼ 11:5Dxðk  NÞ;

N 2 ½1; 4:

ð7Þ

With this process model, a PI controller with transfer function

CðzÞ ¼ K p þ

qi 1  z1

ð8Þ

is designed to obtain a stable output for all possible variations of N. The z variable in this transfer function is not related to a fixed sampling period, but to the number of tiles index k; ðXðzÞ ¼ P1 k k¼0 xðkÞz Þ. The control action is updated every time a tile crosses the outlet temperature sensor position, and this happens at arbitrary instants of time. Therefore, the control action could be updated while a tile is crossing the water spray. This makes a non uniform water layer to be deposited on that tile, that could cause superficial imperfections if the change in the control action is large. In order to avoid those imperfections, the controller should be designed such that the change uðkÞ  uðk  1Þ is low. The idea of the PI design is, then, to track the slow temperature variations, but without trying to track the fast temperature variations, that is, trying to filter out the fast (tile to tile) temperature change (this is because the delay makes it impossible to compensate for the tile to tile change). In order to do so, the PI is designed such that the closed loop bandwidth lies between the cut off frequency of the low frequency part of the inlet temperature, and the first peak of the fast

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Fig. 9. Inlet, outlet and reference temperatures and control action. PI controller.

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frequency part of the inlet temperature, and such that the phase margin is sufficient (larger than 70° for guaranteeing robustness). Looking at Fig. 4, the closed loop bandwidth has been fixed around 0.025 samples1. In order to achieve this, the PI controller is tuned by simulations leading to gains K p ¼ qi ¼ 0:01. As the delay is variant (from 1 to 4), this controller approximately fulfils these conditions for the four possible delays: for delay 1, the bandwidth is 0.02 samples1 and the phase margin 93, while for delay 4, the bandwidth is 0.03 samples1 and the phase margin 73. For the other delays intermediate values are obtained. The experimental results of this PI controller are shown in Fig. 9, where the slow variation of the control action and the outlet average reference temperature tracking can be appreciated. The standard deviation of the tracking error is 3°C. The implementation results show the principal drawback of this strategy, that is the impossibility to detect and correct the arrival of sporadic tiles with a temperature differing more than 5 °C from the mean value. However, in some production lines, the periodic variation between tiles is small with respect to the mean temperature variation of the inlet tiles, and for those lines the PI can be a suitable strategy. 5. Feedforward adaptive control In this section, the highest cost and highest performance strategy is analyzed. Both the inlet and the outlet temperature sensors are used. The easiest control strategy for compensating the measurable disturbance ðT in Þ is an open loop feedforward control. The benefit of the open loop strategy is that only one temperature sensor has to be used and hence, the cost of the control system is lower. The controller (as the process model) is static and simply obtains the control action required for a given inlet temperature and a desired outlet temperature by inverting the identified model (6)

uðkÞ ¼

11:5 : T out; ref  18:38  0:531T in ðkÞ

ð9Þ

The above controller was tested on the plant one day after the identification experiment. In Fig. 10 the inlet, outlet and reference temperatures are shown, together with the control action (flow rate over velocity). The standard deviation of the outlet temperature has been greatly reduced with respect to the non controlled system (3 °C with respect to 7 °C). Nevertheless, the outlet temper-

ature has an average error of about 10 °C. This important deviation from the desired outlet temperature reflects that the process changes with time due to changes in the dryer, in the composition and humidity of tiles, ambient temperature, nozzle wear, and so on, and concludes that an open loop feedforward controller based on a fixed model is not sufficient. The average dc error of the open loop controller could be eliminated by adding an integral feedback term (such as the previous PI controller). Nevertheless the resulting variance would be about 3 °C, that is much larger than that of the identified model error. In order to improve the performance, the process model should be updated continuously, leading to an adaptive control scheme. The online identification algorithm is a standard RLS algorithm [9], that estimates the parameters a; b and c from Eq. (6) based on the inlet temperature, the inverse of the control action and the outlet temperature. The parameter and regression vectors are

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The controller will be the same open loop feedforward controller (9), except that the parameters are updated online by the RLS algorithm. The choice of the forgetting factor is a compromise between rapid identification of fast process changes and the adequate filtering of the very important disturbances. After some simulations, a value of k ¼ 0:995 was chosen for testing the adaptive controller on the plant. In Fig. 11 the inlet, outlet and reference temperatures as well as the control action obtained in the plant experiment are shown, while in Fig. 12 the flow rate and velocity variations are shown. After the initial transient necessary to identify the correct process parameters, the average error is approximately zero, clearly improving the behavior of the fixed controller. The standard deviation equal to 2.2 °C is also lower than that obtained with the fixed controller. In fact it could be even lower because the control action saturates quite frequently in this experiment due to very high inlet temperatures. If the data between tile numbers 50 and 180 are considered (after the RLS initial transient and before the high control saturation interval) the standard deviation is about 1.5 °C, that is, similar to the off line identified model error. In Fig. 13, the evolution of the estimated parameters with time is shown. During the initial transient (first 30 samples where the prediction error is high) the process parameters converge to values

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Fig. 10. Inlet, outlet and reference temperatures and control action. Static feedforward controller.

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Fig. 11. Inlet, outlet and reference temperatures and control action. Adaptive open loop feedforward controller.

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Fig. 14. Trace of the variance–covariance matrix.

sample Fig. 12. Flow rate and velocity variations. Adaptive open loop feedforward controller.

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Fig. 13. Model parameters. Adaptive open loop feedforward controller.

near h  ½24 0:4 11T , that are similar but slightly different from the ones obtained in the off line identification. When the prediction error converges to an independent signal of zero mean (from sample number 35 approximately), the parameters suffer a quite fast and significative change, converging around the values h  ½8 0:6 19T . This is due to the lack of excitation when the process is perfectly controlled. In this case the outlet temperature remains approximately constant at T out  90 C. This means that there is not enough information to identify the surface T out ¼ f ðT in ; 1uÞ ¼ a þ bT in þ c 1u, but only the intersection of this surface with the plane T out ¼ 90 C. As a consequence, the identification algorithm may converge to any surface whose intersection with T out ¼ 90 is the same as that of the true surface. In fact, the previous parameters ð½8 0:6 19T Þ define one of those possible surfaces. It is obvious that in this case the ‘‘true” parameters of the process are not obtained. However, the model is still adequate for control purposes because the predicted outlet temperature is the correct one. The only concern is the possibility of a wind up of the variance–covariance matrix due to this lack of excitation. In Fig. 14, the evolution of the trace of this matrix is shown. From tile number 90 till tile number 180 the trace increases, but not dramatically. From tile number 180 till the end of the experiment, due to the saturation of the actuator, the error is no longer an independent signal of zero mean, and hence, the algorithm has enough

information to identify the ‘‘true” process model (in fact, the parameters converge again to values near the initially identified ones). This leads to the stabilization of the trace (that stops increasing). During normal production, however, there could be long periods of time without saturation and the trace could then grow fast leading to numerical problems. Therefore, for a long term implementation, a trace control strategy should be implemented [11]. A simple trace control strategy is proposed and implemented: the trace of matrix P is saturated to a maximum value of 10. When the trace of P k is larger than 10, it is updated as Pk ¼ Pk  10=traceðP k Þ, in order to avoid excessive growth. Furthermore, when this happens (the trace is saturated), the parameters are not updated. The limit value of 10 for the trace of P was decided after several simulations. The proposed control scheme does not use an explicit feedback term. However, the feedback is implicit in the RLS online identification. The addition of an explicit feedback PI term was tested in simulations, but the performance was not improved. The reason is that the main control problem is the rejection of the inlet temperature disturbance, that is a measurable signal that suffers important changes from one tile to the next. The feedback can only compensate for the slow variations, but this is already done by the online RLS algorithm. The feedback could be interesting if no adaptation is used. In this case, the feedback term would compensate for variations of the process model parameters related to the control action, leading to null mean error. The drawback is that the changes on the parameter related to the inlet temperature are not detected, and hence, the variance of the error is not minimized, leading to a worse performance. 6. Disturbance observer feedforward control In this section an alternative strategy that tends to regulate the slow ”average” and the tile to tile temperature variations using only the outlet pyrometer is addressed. The resulting cost is lower than the adaptive scheme, and the performance is higher than the PI scheme. In this case a disturbance model is identified in order to construct a disturbance observer that predicts the future tile inlet temperatures (or more precisely, a variable related to it), and a specific amount of water is applied to each tile when it reaches the water sprayer. For this reason, the inlet photocell is needed to detect the tiles in this setup. The idea of this controller is to approach the performance of the adaptive feedforward control but avoid the use of the inlet temperature sensor, reducing the implementation cost. The inlet temperature evolution presents a pattern that depends on the size of the tiles and their position inside the drier, as it is shown in Fig. 3. Fig. 4 shows the magnitude of the frequency spectrum of the inlet temperature signal during the identification experiment. High values can be observed at low frequencies (slow

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R. Sanchis, I. Peñarrocha / Journal of Process Control 19 (2009) 1073–1081

temperature variation), and two different peaks at frequencies 0.25 and 0.50 samples1, corresponding with the repetitive temperature pattern of 2 and 4 tiles observed in Fig. 3. The pattern can be identified if model (6) is modified including a disturbance model of the process in the form

T out ðkÞ ¼ a þ bT in ðkÞ þ c

1 1 ! T out ðkÞ ¼ c þ g þ wðkÞ; uðkÞ uðkÞ

ð11Þ

1 where c uðkÞ corresponds with the control action term in (6), and g þ wðkÞ corresponds to the effect of the input disturbance a þ bT in ðkÞ. The constant g represents the average value of a þ bT in ðkÞ along time, and wðkÞ is the disturbance used to model the changes in the inlet temperature. This disturbance is assumed to be the result of filtering a white noise through a Box-Jenkins model, leading to model equation

T out ðkÞ ¼ c

1 þ g þ HðqÞðkÞ; uðkÞ

ð12Þ

where ðkÞ is a white noise and HðzÞ the disturbance filter to be identified ðwðkÞ ¼ HðqÞðkÞÞ. Using the data of the identification experiment (Section 3), g and HðzÞ can be identified off line. The constant g is simply computed as the average of sequence aþ leading to g ¼ 84 C. Using the sequence bT in ðkÞ, wðkÞ ¼ a þ bT in ðkÞ  g as the filter output, and with the help of Ident Matlab toolbox, the filter HðzÞ is estimated to be

HðzÞ ¼

1 þ 0:286z1 þ 0:477z2 þ 0:172z3  0:156z4 ; 1 þ 0:0161z1  0:0142z2 þ 0:0123z3  0:932z4

ð13Þ

whose frequency response is shown in Fig. 15, and whose poles are z ¼ 0:98; 0:99; 0:0008 0:98j. Note that the pole in z  1 estimates the slow variations, the pole in z  1 the variations between consecutive tiles (of frequency 0.5 samples1), and the poles in z  j the variations each 4 tiles (frequency 0.25 samples1). As it was shown in the previous section, the variation of the process parameters along time is significant. An adaptive scheme could then be used, where the gain of the control action ðcÞ, the mean inlet temperature ðgÞ and the inlet temperature variations model ðHðzÞÞ are corrected with each measurement. However, the complexity of the filter to be estimated online makes this approach impractical due to the low robustness of the resulting algorithm. Nevertheless, a fixed realization of the filter HðzÞ can lead to a reasonable performance if a disturbance observer based on this filter is used to estimate the equivalent inlet temperature disturbance. In order to do so, an internal realization of HðzÞ is obtained, leading to

xðk þ 1Þ ¼ AxðkÞ þ BðkÞ; wðkÞ ¼ CxðkÞ þ DðkÞ

ð14Þ

with values

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−1

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0:932 0:0123 0:0142 0:0161 C ¼ ½ 0:776 0:1597 0:4912 0:2699 ;

ð15Þ

1

D ¼ 1:

ð16Þ

The idea is to use the model (14) to predict the future values of ^ þ NÞ with N 2 ½1; 4 for the wðkÞ, i.e. to obtain the estimation wðk considered process. For that purpose, the indirect disturbance measurement wðkÞ, calculated from the real outlet temperature measurement T out ðkÞ,

wðkÞ ¼ T out ðkÞ  g 

c uðkÞ

ð17Þ

is used as the input of a state observer. The equation of the proposed observer is

^xðkÞ ¼ A^xðk  1Þ þ LðwðkÞ  wðkÞÞ ^ ¼ A^xðk  1Þ þ LðwðkÞ  CA^xðk  1ÞÞ;

ð18Þ

where the state is updated with the newest measurement, and the best estimations of ðkÞ and ðk  1Þ are assumed to be zero. With the state estimate, the model can be run in open loop until k þ N (assuming Efðk þ iÞg ¼ 0; i ¼ 1; . . . ; NÞ leading to the disturbance prediction.

^ þ NjkÞ ¼ CAN ^xðkÞ: wðk

ð19Þ

With this prediction, the control action uðk þ NÞ to be applied over the tile that is about to cross the water spray, can be obtained by inverting model (11), leading to equation

uðk þ NÞ ¼

c ; ^ þ NjkÞ T out;ref  g  wðk

ð20Þ

where c ¼ 11:5; g ¼ 84 and T out;ref is the desired outlet temperature. In order to obtain an stable predictor, an adequate vector gain L must be designed, based on the observer dynamics. This is obtained using expressions (18) and (14), leading to

~xðkÞ ¼ ðA  LCAÞ~xðk  1Þ þ ðB  LCBÞðk  1Þ  LDðkÞ; ~ wðkÞ ¼ C ~xðkÞ þ DðkÞ;

ð21Þ

~ ^ where ~xðkÞ ¼ xðkÞ  ^xðkÞ and wðkÞ ¼ wðkÞ  wðkÞ are the state and disturbance estimate errors, respectively. The gain L must guarantee that eigenvalues of ðA  LCAÞ are inside the unit circle, and that the ~ is minimized. As no measurement noise is aseffect of ðkÞ on wðkÞ sumed in the outlet temperature, fast poles are assigned ðz ¼ 0:1; 0:2; 0:4Þ, leading to the gain

L ¼ ½0:2499; 2:3849; 0:3234; 0:9435T :

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−10

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ð22Þ

In Fig. 16, the predictions of T in ðkÞ for N ¼ 4 are compared with the measured temperatures using the gathered data in the identification experiment, showing also the prediction error, resulting in a standard deviation of 4.1 °C. This estimation has been used to simulate the observer based feedforward controller with Eq. (20), leading to the results shown in Fig. 17, where a standard deviation of 2.4 °C was obtained for the tracking error. The pattern of the inlet temperature variations is related to the number of tiles that fit in each dryer plane, that depends on the tile size. The lower the tile size, the higher the period and complexity of the pattern, and hence, the higher the order of the identified filter HðzÞ, making more difficult the computational implementation and algorithm convergence. For very small tile sizes, the temperature variations between tiles are small, and the use of a simple feedback PI controller can be an alternative, as addressed in previ-

R. Sanchis, I. Peñarrocha / Journal of Process Control 19 (2009) 1073–1081 measured (-) and prediced (:) Tin. N=4

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An experimental identification has been carried out, leading to a very simple nonlinear static model that relates the mass of water deposited per tile to the inlet and outlet temperatures. Three different control solutions, that imply different costs and performance, are proposed and tested. The first one (the lowest cost) implies the use of only one temperature sensor at the outlet, and a very simple control algorithm (PI) that only regulates the slow ”average” temperature change, but does not regulate the tile to tile temperature changes. This solution can be valid when the type and size of tiles make the tile to tile difference not too large. The second solution (the highest cost) implies the use of two temperature sensors (at the inlet and outlet) and an adaptive feedforward control algorithm, that regulates not only the slow variation, but also the tile to tile variation. A RLS online identification algorithm was used to update the process model. The performance was very good, with a null average error and approaching the error standard deviation of the process model (about 1.5 °C). This solution is valid for all cases. Finally, a third solution is proposed that approaches the low cost of the PI (only the outlet temperature sensor is used) and the performance of the adaptive scheme. This strategy allows to regulate the slow variation and the tile to tile variation, but with a lower precision than the adaptive control. It is a disturbance observer based feedforward control, where the measurement of the outlet temperature is used to predict the inlet temperature of future tiles (disturbance observer), and this prediction is used to calculate the control action. The performance achieved in simulations with this technique (about 2.4 °C) is in between the PI and the adaptive cases. The main drawback is the need to identify the process and disturbance model every time there is a tile size change in the line. However, these models could be previously calculated and stored in the controller.

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Fig. 17. Simulated behaviour of the observer based feedforward control. Outlet temperature and control action.

This work was supported by the Generalitat Valenciana under project GV00-142-11 and by the Fundació Caixa Castelló-Bancaixa under project P1.1B2007-07. This work was possible thanks to the collaboration of TAU Cerámica. References

ous sections. On the other hand, every time there is a tile size change in the line, the process and disturbance models must be identified and the controller and observer re tuned in order to use the disturbance observer feedforward control. However, on a practical implementation, all the possible models could be previously calculated and stored in the controller. The feedforward adaptive approach can be used in any case, and there is no need to re tune the controller, as the online identification algorithm adapts the controller to the process change, with the only drawback of a higher cost due to the two sensors used. Therefore, the best approach will depend on the size of the tiles and the frequency of tile size changes in the line, as well as the acceptable cost. 7. Conclusions In this paper, the control of the surface temperature of tiles by means of variable water spraying has been studied. First, the experimental setup has been described, including the sensing and actuating procedure.

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