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The objectives of this project are to identify a linear time-invariant dynamical model of an activated sludge process. Such a system is characterized by stiff dynamics, nonlinearities, time-variant parameters, recycles, multivariable with many cross-couplings and wide variations in the inflow and the composition of the incoming wastewater. In this project study, an identification approach based on subspace methods is applied in order to estimate a nominal MIMO state space model around a given operating point, by probing the system in open-loop with multi-level random signals (MRBS). Three subspace algorithms are used, and their performances are compared based on adequate quality criteria, taking into account identification/validation data. As a result, the selected model is a very low-order one, and it describes the complex dynamics of the process well. Important issues concerning the generation of the data set and the estimation of the model order is discussed.
Tujuan dari projek ini adalah untuk mengenalpasti model masa-invariant linear dinamik daripada proses lumpur aktif. Sistem seperti ditandai oleh dinamik kaku, nonlinier, masa-variasi parameter, mengitar semula, multivariabel dengan banyak lintas kopling dan variasi yang luas dalam arus masuk dan komposisi air sisa yang masuk. Dalam kajian ini projek, pendekatan berdasarkan kaedah pengenalan Subspace diterapkan untuk menganggarkan keadaan MIMO space model ukuran dasar sekitar titik operasi tertentu, dengan menyiasat sistem dalam loop terbuka dengan isyarat rawak multi-level (MRBS). Tiga algoritma Subspace yang digunakan, dan prestasi mereka berbanding berdasarkan pada kriteria kualiti yang mencukupi, dengan mempertimbangkan pengenalan akaun / data validasi. Akibatnya, model yang dipilih adalah sangat rendah-order satu, dan itu menggambarkan dinamika kompleks dari proses dengan baik. Isu penting tentang generasi dari himpunan data dan estimasi model urutan dibahas.
TABLE OF CONTENTS
CHAPTER TITLE PAGE
List of Tables
LIST IF FIGURES
List of Abbreviation
List of Symbols
List of Appendices
1.1 Background of Study
The activated sludge process is defined as a system in which a large number of biological organisms are maintained and continuously circulated to be in constant contact with the organic wastewater in the presence of air. The bio process is principally constituted by two sequential tanks, an aerator and a settler. The bacteria and other microorganisms feed on the organic matter constituent of the incoming wastewater, thereby reducing the strength of the waste. The sludge is separated from the mixed liquor in the solids separator (settler). The arranged sludge portion is reprocessed as return activated sludge from the settler to the aeration reactor so that the micro-organisms content in the reactor is maintained at the reaction sustenance level. Excess sludge, which is not recycled, is extracted from the system as waste activated sludge and consequently processed for deferent uses.
1.2 Problem Statement
There are a lot of system identification techniques, the most traditional system identification techniques are the prediction error method (PEM) and the instrumental variable method (IVM).
Computing the PEM model can sometimes be very difficult.
The IVM attempts to deliver parameter estimates by only solving linear equation systems.
The objective of this project includes three subjects.
Design the input excitation using multilevel PRBS signal.
Apply subspace algorithm;
Compare the performance of the three methods of the subspace algorithms
1.4 Scope of Study
This study focus on finding the best model for activated sludge process, for this approach, three methods of the subspace algorithms are used.
MOESP - Multivariable Output-Error State-space model identification.
N4SID - Numerical algorithm for Subspace State Space System Identification.
''robust'' N4SID - Robust Numerical algorithm for Subspace State Space System Identification.
1.5 Importance of Study
The identification techniques convert a non-linear system to a linear time invariant (LTI) system, therefore the identified system can use easily for further approaches such as computer estimation instead of doing experimental act on the real system or designing controllers for the system because designing controller such as PI or PID for the linear system is easier than to a non-linear systems.
2 LITERATURE REVIEW
2.1 History of and Introduction to Wastewater Treatment
In the industrialized western society, wastewater treatment has become a prerequisite to maintain satisfactory surface water quality. This is largely a consequence of the historical development through which sewers have become the predominant sanitation system, transporting waste to surface at water. In the 19th century, industrialisation and urbanization gave rise to a dramatic increase in population density and lack of proper sanitation led to several epidemics in different countries. After the Great Plague in London in 1857, the British government decided to use water to transport the pathogen containing waste from the cities to the sea, with the introduction of water closets and sewers  ('seawards'). In the decades after the commencement of the London sewer procedure, in many western metropolises sewer systems were built and suited favoured over additional competing sanitation systems, such as the Liernur vacuum system in some Dutch and German metropolises, and barrel accumulation systems ,for instance, in Sweden, Germany and The Netherlands (Oremus 1990, Lange and Otterpohl, 1997).
The intensified pollution load to surface at water following the institution of sewers led to an extreme deterioration of water quality in numerous surface waters. In Table 2.1, the most significant groups of pollutants in local sewage are designated (Metcalf & Eddy, 1991), composed with effects on surface water quality, and an indication of the connected time scale of the effects (Lijklema et al., 1993, Schilling et al., 1997). It should be noted that this table applies to domestic wastewater. Significant industrial discharges into the sewer may have a strong influence on wastewater composition, while the nature of the pollutants is very industry specific. For industrial wastewater, source control is the preferred solution, and industrial wastewater is not considered further in this thesis.
Table Table 2.1: Pollutants in domestic wastewater and effects
In the first half of the 20th century, many rivers, streams and other surface waters were so deeply dirty that they could be qualified as stinking open sewers (Dirkzwager and Kiestra, 1995). Mainly acute problems from solids and organic material became manifest. The naturally occurring ecosystems had been seriously distorted and in many cases normal aquatic life had almost disappeared as a result of oxygen depletion due to biological oxidation of organic compounds. In addition, stench resulting from anaerobic conditions was frequently a problem. It took until the second half of the 20th century before legislation on wastewater treatment became effective in most western countries. The focus was initially on reduction of acute problems, especially the reduction of oxygen depletion problems through removal of biodegradable organic compounds, and, to a lesser extent, ammonia. In The Netherlands, the Law on Surface Water Pollution (WVO) became effective in 1970.
1 Figure 2.1 Installed sewage treatment capacity in The Netherlands. In 1970, the WVO became effective (Dirkzwager and Kiestra, 1995). Biological treatment is applied most frequently. (Capacity is expressed as M p.e., millions of person equivalents)
Because of legislation, over the past few decades wastewater treatment has been introduced on a large scale. In terms of treated raw material, wastewater treatment can now be considered the largest process industry, with an average flow of approximately 40*being treated daily in Western Europe (Vanrolleghem, 1994). Sewage treatment is accomplished most economically by biological oxidation, and the activated sludge process with its variants is the most commonly used process. Figure 2.1 illustrates the progress of different treatment technologies in The Netherlands over the past decades, which shows the predominance of biological treatment.
1 Figure 2.2 Archetypal flow scheme of conventional activated sludge plant.
Figure 2.2 shows an archetypal flow scheme of an activated sludge plant. In the aeration tank, air is mechanically imparted, and different types of micro-organisms in the so-called activated sludge oxidise pollutants to less harmful compounds whilst producing new biomass. So-called heterotrophic bacteria oxidise carbonaceous organic compounds to carbon dioxide and autotrophic bacteria partially oxidise ammonia to nitrate, a process referred to as nitrification. After biological treatment, the activated sludge is separated from the wastewater in the final clarifier. The treated wastewater is disposed into the surface water and the sludge is partly recirculated to the activated sludge tank and partly wasted as excess sludge.
The process scheme depicted in Figure 2.2 shows only the biological treatment of a typical wastewater treatment plant. In general, wastewater treatment also includes mechanical treatment to remove floating and settleable solids as a first step and other operations such as sludge treatment and chemical treatment.
2.2 Activated sludge Process model
Figure 2.3 shows the schematic of the activated sludge process that is described by four non-linear equations.
1 Figure 2.3 Activated Sludge process model
Where the state variables, X (t), S (t), C (t) and Xr(t), represents the concentrations of biomass, substrate, dissolved oxygen and recycled biomass respectively. D (t) is the dilution rate, while Sin and Cin correspond to the substrate and dissolved oxygen concentrations of influent stream. The parameters r and Î² represents the ratio of recycled and waste flow to the influent flow rate, respectively. The kinetics of the cell mass production is defined in terms of the specific growth rate µ and the yield of cell mass Y. The term Ko is a constant. Cs and KLadenote the maximum dissolved oxygen concentration and the oxygen mass transfer coefficient, respectively. The Monod equation gives the growth rate related to the maximum growth rate, to the substrate concentration, and to the dissolved oxygen concentration:
Where µmax is the maximum specific growth rate, Ks is the affinity constant and Kc is the saturation constant.
2.3 Identification Methods
There are some methods for identifying a system, some of them are as follows:
Prediction error method (PEM)
Instrumental variable method (IVM)
Prediction error Method and Instrumental variable method are the most traditional methods. They have some problem such as PEM model can sometimes be extremely difficult. In general, a multi-dimensional nonlinear optimization problem must be solved, and Instrumental variable method attempts to hand over parameter estimates by just working out linear equation systems. Nevertheless, the use of these models is fairly heavy in the general multivariable case, and the numerical reliability may be undesirably high for complicated cases involving large system orders and many outputs (Viberg, 2002).
Subspace-based system identification approach is a branch that has been newly developed in system identification (about20 years old at present), that has attracted much attention, owing to its computational simplicity and creativeness in identifying dynamic state space linear multivariable systems. These algorithms are numerically robust and do not involve nonlinear optimization techniques, i.e. theyare fast (no iterative) and accurate (since no problems with local minima occur). The computational complexity is modest compared to PEM, inputs and outputs are large. Because programs of large dimensions are usually found in the process industry, subspace identification methods are very promising in this scope.Thus, a large number of successful applications of subspace identification methods for simulated and real processes have been reported in the literature.
Subspace-based methods for state-space modeling have their origin in the state-space realization, as advanced by Ho and Kalman (1966). These ways determinea statespace model from a given impulse response, which received a massive attention in the signal processing domain in the late 1970s. In the system identification area, one normally has available input-output data rather than the measured impulse response. In this subject, subspace methods were developed in the late 1980s.
The term ''subspace identification method'' was introduced by Verhaegen and Deprettere (1991). There are now many various versions of subspace approaches. These include an early version of the subspace algorithm presented in the paper by Moonen, De Moor, Vandenberghe, and VandeWalle (1989), Canonical Variate Analysis (CVA) by Larimore (1983, 1990), Multivariable Output-Error Statespace model identification(MOESP) by Verhaegen and Dewilde (1992), InstrumentalVariableSubspace-based State-Space SystemIDentification (N4SID) by Ottersten and Viberg(1994), Numerical algorithm for Subspace State Space System IDentification (N4SID) by Van OverscheeandDe Moor (1994, 1996), Canonical Correlation Analysis (CCA) by Peternell, Scherrer, and Deistler (1996) and Deterministic and Stochastic subspace system identification and Realization (DSR) by Di Ruscio(1997).
All subspace identification patternsinclude of three principal steps: estimating the predictable subspace for multiple future steps, then extracting state variables from this subspace and lastly fitting the estimated states to a state-space model. However, each subspace identification approach looks fairly different from others in goal, computational tools and interpretation. The main differences among these subspace identification methods lie in the regression or projection methods used in the first step to remove the effects of the future inputs on the future outputs and, thereby, estimate the predictable subspace, and in the invisible variable methods used in the second step to extract estimates of the states. A general overview of the new in subspace identification methods is presented in De Moor, Van Overschee, and Favoreel (1999) and Favoreel, DeMoor, and Van Overschee (2000).
The mainbenefits of these algorithms are that they just need input-output data and very little earlier knowledge about the system. In total, these algorithms are based on system theory, geometry and numerically stable no iterative linear algebra operations, such as QR (or LQ) factorization, SVD (singular value decomposition) and its generalizations, for which valid numerical tools are well-known (Golub&VanLoan, 1996). A disadvantage against subspace identification method is that the physical knowledge of the process, in the obtained model, is lost, which is a characteristic of black-box patterns. For example, the states are ''artificial'' and it is impossibleto understand how a process variable, which is not directly contained in the model, affects the process. Moreover, a large amount of data is needed to obtain accurate models. In fact, generating and collecting data of some processes can be overpriced. Important issues involved in developing a model through subspace identification approaches can be found in Amirthalingam and Lee (1999).
Subspace identification methods have latelyobtained a certain level of perfection. Some of t subspace identification algorithms are:
CCA: unconstrained CCA algorithm (uCCA), essentially the same CVA algorithm proposed by Larimore (1983); and constrained CCA algorithm (cCCA), proposed by Peternell et al. (1996).
MOESP: refined version of the past outputs (PO) scheme of the MOESP algorithm in the SMI Toolbox by Haverkamp and Verhaegen (1997).
N4SID: N4SID function (n4sid.m) in the MATLAB System Identification Toolbox v.4.0.4 (Ljung, 1997), that implements the N4SID algorithm from Van Overschee and De Moor (1994) and the ''robust'' N4SID algorithm from Van Overschee and De Moor (1996).
DSR: DSR algorithm in the D-SR Toolbox by Di Ruscio (1997).
In this project the activated sludge process system with three of these algorithms are identified, and the performance of each is compared to another algorithms.
This topic covers the activities that should be carrying out in order to achieve the objectives of the study. The details of the activities are shown in Figure 3.1.
First, the problem is identified. This is important so that the objectives of the tests are stated clearly and can be reached after the tests are carried out. After that, the literature reviews are collected from a lot of sources such as books, journals and reports. Then, modeling of the system by using SIMULINK is carried out. The data from simulations are collected while the modeling is carrying on and the data is analyzed based on appropriate performance quality criteria to choose the best model for the system.
The diagram shows the stages of the process in identification of Activated sludge process. The process consists of: Literature review, Building models, Identification based-on the subspace algorithms, comparing the performance of those three methods, Data evaluation and Documentation.
1 Figure 3.1 Flow chart of the activities in the study
3.2 Collecting Literature Reviews
Before this research is done, it is very important to know the theories related with the field research, for example the basic knowledge of Activated sludge, literature reviews are done before data collection is carried out.
Literature reviews are done through reading education materials such as books and journals as in Chapter 2.In the literature review, some overview of Activated sludge process, subspace algorithms are fully discussed, that lists of books, resources and articles are used.
3.3 Building models of the Activated sludge process
The model of activated sludge process is shown in figureâ€¦..is described by four non-linear differential equations. SIMULINK is used to model based on the equations.
In figureâ€¦is shown the subsystem with inputs and output of the activated sludge process.
Figureâ€¦ shows the inside of the subsystem.
Each function those are used in this model has their function calculation as an example the function of is shown in figureâ€¦.
3.4 Data collection
For Identifying a system with a sub-space algorithms need to have data about the inputs and outputs the system as it mentioned in literature reviews these algorithms works on Input and output data and little earlier knowledge about the system.
Tableâ€¦..shows the initial values of the system.
Tableâ€¦.shows the parameters value.
3.5 Identifying the system with subspace algorithms
In this part the system will be identified based on the data from the previous part in three deferent ways, and finally, the performance of the results are compared to each other, and the best one is chosen.
A linear time-invariant system with inputs uk, outputs yk and states xk, described by four matrices A, B, C and D. The symbol âˆ† represents a delay.
Note the inherent feedback via the matrix A (representing the dynamics). In the deterministic identification problem, the circled signals (input uk and output yk) are known. The state is unknown, but will be determined as an intermediate result in the subspace identification algorithms.
Figure 2.2 state space model diagrams
3.5.1 Block Hankel matrices and state sequences
In this Subsection we introduce the notation for block Hankel matrices and for system related matrices. Block Hankel matrices play an important role in subspace identification algorithms. These matrices can be easily constructed from the given input-output data. Input block Hankel matrices are defined as:
The number of block rows (ii) is a user-defined index which is large enough i.e. it should at least be larger than the maximum order of the system one wants to identify (see below). Note that, since each block row contains m (number of inputs) rows, the matrix Uo|2i-1 consists of 2mi rows.
The number of columns (jj) is typically equal to s-2i+1 which implies that all given data samples are used. Throughout the book, for statistical reasons (see also Subsection 1.4.4), we will often assume that j,sà âˆžfor deterministic (noiseless) systems this will not be necessary.
The subscripts of U0|2i-1 , U0|i-1, U0|i denote the subscript of the first and last element of the first column in the block Hankel matrix. The subscript "p" stands for "past" and the subscript "f" for "future". The matrices Up (the past inputs) and Uf (the future inputs) are defined by splitting U0|2i-1 in two equal parts of i I block rows. The matrices Up+ and Up- f on the other hand are defined by shifting the border between past and future one block row down_.
Note that the distinction between past and future is somewhat loose, since both the matrices Up p and Up+ p are denoted by "past inputs". These loose notations are however useful when explaining concepts intuitively. Note also that the past and future inputs have many elements in common. For instance the input ui i can be found in Up as in Uf f However, the corresponding columns of UP and Uf have no elements in common, and thus the distinction between past and future.
The output block Hankel matrices Y0|2i-1 , YP , Yf , YP+ , Yf- f are defined in a similar way. we define the block Hankel matrices consisting of inputs and outputs as W0|i-1:
Similarity as before is defined as:
State sequences play an important role in the derivation and interpretation of subspace indetification algorithms. The state sequence is defined as:
Where the subscript i denotes the subscript of the first element of the state sequence. Anaalogous to the past inputs and outputs, we denote the past state sequence by :
System related matrices
Subspace identification algorithms make extensive use of observability and controllability matrices and of their structure. The extended (i > n) observability matrix (where the subscripts i denote the number of block rows) is defined as:
We assume the pair to be observable, which implies (see for instance) that, the rank of is equal to n. The reserved extended controllability matrix (where the subscripts i denote the number of block columns) is defined as:
We assume the pair to be controllable. The controllable modes can be either stable or unstable. The lower block triangular Toeplitz matrix is defined as:
Matrix input-output equations
The following Theorem states how the linear state space relations of formula (2.1)-(2.2) can be reformulated in a matrix form. The Theorem was introduced above and is very useful in many proofs of, and insights in subspace identification algorithms.
Theorem 1 Matrix input-output equations
The proof follows directly from the state space equations (2.1)-(2.2). The geometric interpretation of equation (2.5) is illustrated in Figure â€¦.
Before stating the main deterministic identification Theorem, the following remark that emphasizes the symmetry between the different Chapters is in order: For each of the separate identification problems (Chapter 2, 3 and 4) we present a main Theorem which states how the state sequence and the extended observability matrix can be extracted from the given input-output data. After having treated the three Theorems for the three different cases (deterministic, stochastic and combined deterministic-stochastic identification), it will become clear that they are very similar. These similarities will be treated in Section 4.5. We mention this fact early in the book, before the Theorems are introduced, so that the synthesis in Section 4.5 will be anticipated by the attentive reader.
The consequences of the deterministic identification Theorem are twofold:
The state sequence can be determined directly from the given data and without knowledge of the system matrices A, B, C, D.
The extended observability matrix can be determined directly from the given input-output data.
We will then describe how the system matrices A, B, C, D can be extracted from these intermediate results and . An overview of the overall identification procedure is presented in figure â€¦.
In the main identification Theorem, we introduce two weighting matrices . Suffices to state here that specific choices of the matrices lead to different identification algorithms of the literature and that the choice of the weights determines the state space basis in which the final model is obtained.