Çimento Harmanlama Prosesinin Kalman Kestirimcisi İle Tanılanması

Abstract

Gelişen endüstrinin ihtiyaçlarında, sürekli verimlilik, zaman, maliyet, v.s. gibi faktörlerin daha iyi değerlendirilmesi, gün geçtikçe önemini artırmaktadır. Bu bağlamda otomasyon da daha çok önem kazanarak sürekli yeni uygulama alanlarında yer almaktadır. Bu ihtiyaçların büyük bir bölümünü gösteren çimento sektörü de bu uygulamalar arasında yer alarak kendine otomasyondan önemli bir pay edinmekte ve otomasyonun en etkin yollarından biri olarak değerlendirilen Kalman filtresi, bu sektörde de önemli bir rol almaktadır. Bölüm l'de çimento prosesine kısa bir giriş yapılarak ardından Kalman filtresine de kısa bir tanıtım yapılmakta, Kalman tarafından ortaya konulan ve bilgisayarla uygulanmasına yönelik ayrık model tanıtılmaktadır. Bölüm 2'de çimento sektöründen örnek gösterilen bölüm geçmektedir. Bu bölümde çimento üretiminin ana yapısını oluşturan iki ana prosesten biri olan, harmanlama prosesinin, ele alınması ve bunun üzerinde Kalman filtresinin ham madde karışımının oksit konsantrasyonlarının tahmin ve filtrelenmesi detaylı olarak ele alınmaktadır. Burada alman karışım numunelerinin X-ışını floresans analizi ile oksit konsantrasyonları analiz edilerek, proses ölçüm değerleri olarak Kalman filtresine beslenmekte ve burada filtre edilerek tahminler sonucu ham madde durumları verilmektedir. Bölüm 3 'de Kalman filtresinin geniş tanıtımı yer almaktadır ve bunun ayrık ve sürekli olan iki modelinin oluşturulması yer almaktadır. Bu iki yapının iskeletini oluşturan ana denklemlerde özet tablolarla verilmektedir. Bölüm 4 'de NUH Çimento harmanlama prosesinin ölçüm verileri üzerinde, MATLAB programının kullanımı ile, proses durum modelinin tahmini için bir Kalman estimatörü uygulaması ve bunun değerlendirilmesi olarak çeşitli Qk ve Rk değerlerinin sonuçlan verilmektedir. Sonuç olarak, bu çalışmanın bütünü, Kalman filtresinin önemini vurgulamakta ve bunun sağladığı olanakların kapsamını çimento sektöründe belli bir boyutta yansıtmaktadır. Bu tezin konusu oldukça kapsamlı olmakla birlikte Bölüm 4'te yapılan bilgisayar programı ile sadece belirli temel çerçeveler ele alınmıştır. Ancak bu çerçevenin çok daha geniş boyutlara getirilebileceği de unutulmamalıdır. Bu tezin gösterebileceği ana fikir, Kalman'ın önemi ve bugünün endüstriyel teknolojisinde sağlayabileceği önemli gelişmelerin kapsamıdır.

After the increasing needs in the field of cement production and specially at the time where costs in general are the factor to be mainly considered. Thus, researches towards developing techniques for the automation of processes gained more increasing speed. Cement raw material blending, is one of the two main processes in cement production which has become a field of studies and developments of new techniques in process control. A cement raw material blending and grinding operation was modelled using the state variable method. A time-varying Kalman filter was applied to obtain estimates of the input feedstream concentrations from X-ray flourescence measurements. The requirement for the observability of the feedstream concentrations was found to be a linearly independent set of flowrate time histories. The estimation algorithm was evaluated by application to real cement plant data. In Chapter 2, such process is presented form an example plant with real process data consideration. Following a Kalman filter is detailed in Chapter 3 by its discrete and continuous forms. Such a filter's diagram is shown in Figure 0. 1. Figure 0. 1 The Kalman filter in block diagram One definition, a Kalman filter is "an estimator for what is called the linear- quadratic-Gaussian problem, which is the problem of estimating the instantaneous state of a linear dynamic system perturbed by Gaussian white noise-by using measurements linearly related to the state, but corrupted by Gaussian white noise. This type of estimator is statistically optimal with respect to any quadratic function of the estimation error. " The Kalman filter is the general solution to the recursive, minimised mean square estimation problem within the class of linear estimators. The Kalman filter gives a linear, unbiased, and minimum error variance recursive algorithm to estimate the unknown states of a dynamic process from noisy data taken at discrete real-time intervals. States, in this context, refer to any quantities of interest involved in the dynamic process, e.g. position velocity, chemical concentration, etc. For Gaussian random variables the Kalman filter is the optimal linear predictor-estimator and for variables of forms other than Gaussian the estimator is the best only within the class of linear estimators. The filter requires a knowledge of the second-order statistics of the noise of process being observed and of the measurement noise in order to provide the solution that minimises the mean square error between the true state and the estimated state. Kalman filtering provides a convenient means of determining the weightings (denoted as gains) to be given to input measurement data. It also provides an estimate of the estimated state's error statistics through a covariance matrix. Hence the Kalman filter chooses the gain sequence and estimates the estimated state's accuracy in accordance with the variations (in terms of accuracy and update rate) of input data and modelled process dynamics. It should be noted that the quality of the estimation, as described through the error covariance matrix can in many cases be determined a priori, and would therefore be independent of the observations made. The Kalman filter has been used extensively for many diverse applications. For example, Kalman filtering has proved useful in navigational and guidance systems, radar tracking sonar ranging, and satellite orbit determination. Here the main concern is the derivation of the Kalman filter algorithm from the point of it being a linear observer, and with how the filter algorithm may be used in practice. As the Kalman filter is generally implemented on digital computers the discrete time form of the algorithm is mostly the considered one. The Discrete Kalman Filter Algorithm The Kalman filter estimates a process by using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of (noisy) measurements. As such the equations for the Kalman filter fall into two groups. These are; i) time update equations ii) measurement update equations The time update equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The measurement update equations are responsible for the feedback i.e. for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The time update equations can also be thought of as predictor equations, while the measurement update equations can be thought of as corrector equations. Indeed the final estimation algorithm resembles that of a predictor-corrector algorithm for solving numerical problems. The specific equations for the time and measurement updates are presented below in Table 0.1. Table 0.1 Summary of Discrete Kalman Filter Equations From Table 0.1 notice how the time update equations project the state and covariance estimates from the time step k to step k+1. The first task during the measurement update is to compute the Kalman gain matrix, Kk. The next step is to actually measure the process to obtain Zk, and then to generate an a posteriori state estimate by incorporating the measurement as in the state estimate update equation from Table 0.1. The final step is to obtain an a posteriori error covariance estimate via error covariance update equation given from Table 0.1. After each time and measurement update pair, the process is repeated with the previous a posteriori estimates used to project or predict the new a priori estimates. This recursive nature is one of the very appealing features of the Kalman filter, it makes practical implementations much more feasible (for example) an implementation of a Wiener filter which is designed to operate on all of the data directly for each estimate. The Kalman filter instead recursively conditions the current estimate on all of the past measurements. Such a recursive iteration procedure is shown in figure 0.2 below. Initial parameter and error estimation Compute Kalman gain Project ahead for parameters and error Update estimate with measurement Compute error covariance for updated estimate Figure 0.2 Kalman filter loop Filter Parameters and Tuning In the actual implementation of the filter, each of measurement error covariance matrix Rk and the process noise Qk might be measured prior to operation of the filter. In the case of the measurement error covariance Rk in particular this makes sense because we need to be able to measure the process (while operating the filter) we should generally be able to take some off-line sample measurements in order to determine the variance of the measurement error In the case of Qk, often times the choice is less deterministic. For example, this noise source is often used to represent the uncertainty in the process model. Sometimes a very poor model can be used simply by "injecting" enough uncertainty via the selection of Qk. Certainly in this case one would hope that the measurement of the process would be reliable. In either case, whether or not we have a rational basis for choosing the parameters, often times superior filter performance (statistically speaking) can be obtained by "tuning" the filter parameters Qk and Rk. The tuning is usually performed off-line, frequently with the help of another (distinct) Kalman filter. In closing we note that under conditions where Qk and Rk are constant, both the estimation error covariance Pk and the Kalman gain Kk will stabilize quickly and then remain constant. If this is the case, these parameters can be pre-computed by either running the filter off-line, or for example by solving error covariance extrapolation equation for the steady-state value of Pk by defining Pk (-) = Pk (+) and solving for Pk. It is frequently the case however that the measurement erroro (in particular) does not remain constant. For example, when sighting beacons in our optoelectronic tracker ceiling panels, the noise in measurements of nearby beacons will be smaller than that in far-away beacons. Also, the process noise Qk is sometimes changed dynamically during filter operation in order to adjust to different dynamics. For example, in the case of tracking the head of a user of a 3D virtual environment we might reduce the magnitude of Qk if the user seems to be moving slowly, and increase the magnitude if the dynamics start changing rapidly. In such a case Qk can be used to model not only the uncertainty in the model, but also the uncertainty of the user's intentions. Following the above, in Chapter 4 a MATLAB program application is presented which goals to form a Kalman estimator on a cement blending process that optimises oxide concentrations of the raw meal mixture. Here the process is evaluated for four oxide variations with basic input of three raw material feedstream states. As the data obtained from NUH Cement Factory (which is the example plant where the blending process is taken into account), the measurements are evaluated in the estimator to obtain a model for the process. It is reasonable to say that these measurements contain the measurement noise. Resulting diagrams follow the program and show the effect of changing Rk and Qk on the performance of the estimator. As for the conclusions it is possible to see that a rapid convergence and reasonable disturbance rejection were occurred. The results show that as the measurement error covariance Rk approaches zero, the actual measurement Zk is "trusted" more and more, while the predicted measurement Hk xk (-) is trusted less and less. On the other hand, as the a priori estimate error covariance Pk(-) approaches zero the actual measurement a is trusted less and less, while the predicted measurement HkXk (-) is trusted more and more.