E. Noldus

Ghent University, Gent, VLG, Belgium

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Publications (30)22.68 Total impact

  • M Sbarciog, M Loccufier, E Noldus
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    ABSTRACT: Microorganisms growth processes are encountered in many biotechnological applications. For an increased economic benefit, optimizing their productivity is of great interest. Often the growth is inhibited by the presence in excess of other components. Inhibition determines the occurrence of multiple equilibrium points, which makes the optimal steady state reachable only from a small region of the system state space. Thus dynamic control is needed to drive the system from an initial state (characterized by a low concentration of microorganisms) to the optimal steady state. The strategy presented in this paper relies on the solutions of two optimization problems: the problem of optimal operation for maximum productivity in steady state (steady state optimization) and the problem of the start-up to the optimal steady state (transient optimization). Steady state optimization means determining the optimal equilibrium point (the amount of microorganisms harvested is maximum). The transient optimization is solved using the maximum principle of Pontryagin. The proposed control law, which drives the bioreactor from an initial state to the optimal steady state while maximizing the productivity, consists of switching the manipulated variable (dilution rate) from the minimum to the maximum value and then to the optimal value at well defined instants. This control law substantially increases the stability region of the optimal equilibrium point. Aside its efficiency, the strategy is also characterized by simplicity, being thus appropriate for implementation in real-life systems. Another important advantage is its generality: this technique may be applied to any microorganisms growth process which involves only one biochemical reaction. This means that the sequence of the control levels does not depend on the structure and parameters of the reaction kinetics, the values of the yield coefficients or the number of components in the bioreactor.
    Computer methods and programs in biomedicine 11/2011; 104(2):112-9. · 1.56 Impact Factor
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    ABSTRACT: This study analyses the steady state behaviour of biological conversion systems with general kinetics, in which two consecutive reactions are carried out by two groups of micro-organisms. The model considered is a realistic description of wastewater treatment processes. A step-wise procedure is followed to reveal the mechanisms affecting the occurrence of steady states in terms of the process input variables. It is clearly demonstrated how taking into account inhibition effects by simply including additional inhibition terms to the kinetic expressions, a common practice, influences the model's long term behaviour. The overall steady state behaviour of the model has been summarized in easy-to-interpret operating diagrams, depicting the occurrence of steady states in terms of the reactor dilution rate and the influent substrate concentration, with well-defined boundaries between distinct operating regions. This knowledge is crucial for modelers as steady state multiplicity--in the sense that more than one steady state can be reached depending on the initial conditions--may remain undetected during simulation. The obtained results may also serve for experimental design and for model validation based on experimental findings.
    Mathematical biosciences 09/2010; 228(2):160-70. · 1.30 Impact Factor
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    ABSTRACT: The operation of biochemical reaction systems requires substantial expertise and a good understanding of system dynamics. Due to various inhibition e�ects, these systems usually possess several equilibrium points. Selecting the initial state of the system is as important as selecting the system's inputs for a proper operation and for achieving the technological goal. This paper presents an anaerobic digestion system with two biochemical reactions, which possesses six equilibria for some ranges of system's inputs. The system's stability boundary separates the set of initial states from which the system converges to the desired operating point and the set of initial states which lead the system to a totally undesired condition, the acidi�cation point. A methodology for estimating this stability boundary is described. Additionally , a procedure for visualizing the extent of the estimated boundary in a lower dimensional space is introduced. The algorithms are simple and provide accurate estimates. Moreover, the results may be displayed as two-dimensional diagrams that can be easily understood by the system's operator.
    11th Computer Applications in Biotechnology, Leuven, Belgium; 07/2010
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    ABSTRACT: Finding appropriate operating strategies for biochemical reactors is a difficult task. One of the main reasons is the complexity of the underlying biological processes. This paper presents a methodology for determining simple operating strategies for anaerobic digestion systems characterized by a two populations model. These strategies are the outcome of a thorough analysis of system’s dynamics correlated with the technological interpretation of the results. The analysis is performed without considering a precise mathematical expression of the bacteriological growth functions. Regions in the inputs space are determined for which various numbers of equilibrium points are obtained. Their location in state space and their local stability properties are assessed. Qualitative information on the technological significance of the equilibrium points is provided. The obtained results and conclusions allow to select appropriate inputs and initial states to achieve a good operation of the process.
    Biochemical Engineering Journal. 01/2010;
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    ABSTRACT: This paper investigates the influence of parameter uncertainty in the reaction rate function on the efficiency of a bang‐bang control strategy for a microorganism’s growth process. The bang‐bang controller is developed to drive the bioreactor from an initial state to a small neighbourhood (target set) of the optimal steady state, while maximizing the productivity. Once the target set is reached, the control effort is switched to the optimal dilution rate, which allows the process to converge to the optimum. This simple control strategy uses a model of the process i) to determine when the switching in the control effort must occur, ii) to determine which initial states will lead the system to an optimal steady state, and iii) to predict the optimal equilibrium point. For certain kinetic coefficients of the model a wide region in the process’s kinetic coefficients space is identified, for which the bang‐bang control will perform reasonably. Moreover, the productivity of the process will be higher than the one predicted by the model in almost the entire robustness region.
    AIP Conference Proceedings. 10/2008; 1051(1):401-412.
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    ABSTRACT: This paper deals with the optimization of biochemical reaction systems of rank one. Two optimization problems are solved: the problem of optimal operation for maximum productivity in steady state and the problem of the start-up to the optimal steady state. Application of Pontryagin's maximum principle shows that the controller is of the bang–bang type, with no singular intervals. The determination of the optimal switching surface involves the solution of a two point boundary value problem. Solving such a difficult problem is avoided by choosing candidate switching surfaces on a heuristic basis. This study shows that switching on the stability boundary of the nominal operating point corresponding to the maximum dilution rate is the best choice. Here the value of the cost index is minimum amongst the various switching surfaces considered and the stability boundary satisfies the conditions imposed on a candidate switching surface for proper operation. Simple, robust algorithms are formulated for accurately estimating the system's stability boundary. The obtained results display the influence of feedback control on the stability of the set point. The bang–bang controller substantially increases the set point's region of attraction in state space as compared to the uncontrolled bioreactor.
    International Journal of Control 05/2008; 81(5):836-850. · 1.01 Impact Factor
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    ABSTRACT: The stability of a class of biochemical processes defined by a set of m biochemical reactions involving n components is analysed. The processes operate in a continuous mode and possess at least two stable equilibrium states: the normal operating point and a biological wash out state. Using a canonical state representation of the process dynamics the geometric structure of the operating point's stability boundary is characterized. Numerical algorithms are developed to evaluate this boundary and to visualize its extent in state space. The proposed technique is illustrated with a representative engineering example.
    Journal of Computational and Applied Mathematics 01/2008; 215(2):557-567. · 0.99 Impact Factor
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    ABSTRACT: In this paper, the influence of microbial growth kinetics on the number and the stability of steady states for a nitrogen removal process is addressed. A two-step nitrification model is studied, in which the maximum growth rate of ammonium oxidizers is larger than the one of nitrite oxidizers. This model describes the behavior of a SHARON reactor for the treatment of wastewater streams with high ammonium concentrations. Steady states are identified through direct calculation using a canonical state space model representation, for several types of microbial kinetics. The stability of the steady states is assessed and the corresponding phase portraits are analyzed. Practical operation of a SHARON reactor aims at reaching ammonium conversion to nitrite while suppressing further conversion to nitrate. Regions in the input space are identified that result in this desired behavior, with only nitrite formation. It is demonstrated that not only the dilution rate plays a role, as is commonly known, but also the influent ammonium concentration. Besides, the type of microbial (inhibition) kinetics has a nonnegligible influence. While the results indicate that product inhibition does not affect the number of steady states of a (bio)reactor model, it is shown that substrate inhibition clearly yields additional steady states. Particular attention is devoted to the physical interpretation of these phenomena.
    Biotechnology and Bioengineering 12/2007; 98(4):882-93. · 3.65 Impact Factor
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    ABSTRACT: This contribution deals with the behaviour of a SHARON reactor for nitrogen removal from wastewater streams with high ammonium concentrations. A system analysis is performed on a two-step nitrification model, describing the behaviour of such a reactor. Steady states are identified through direct calculation using a canonical state space model representation. Practical operation of a SHARON reactor aims at reaching ammonium conversion to nitrite only (nitritation), while suppressing further conversion to nitrate. It is shown how this desired behaviour can be obtained by setting the dilution rate dependent on the influent ammonium concentration. The impact of microbial growth characteristics on the suitable operating region is examined, as well as the effect of reactor temperature and pH. Advice is given for robust reactor design and operation.
    Water Science & Technology 02/2007; 56(6):145-54. · 1.10 Impact Factor
  • E. Noldus
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    ABSTRACT: A sufficient condition for the existence of periodic motion is derived, for a class of autonomous, nonlinear feedback systems. The criterion is stated in terms of the frequency response of the linear part of the system, and the parameters, characterizing the nonlinearity. In this way, a relationship is established, with commonly used frequency domain theorems, in the stability analysis of the same type of systems.Eine wichtige Aufgabe in der nichtlinearen Theorie der selbsttätigen Regelung bilden die bei der Untersuchung von Selbstschwingungen in Regelungssystemen auftretenden Fragen. Ein Frequenzkriterium für die Existenz periodischer Bewegungen eines Regelkreises wird angegeben, bei dem die Untersuchung des geschlossenen Regelkreises auf die Untersuchung der Ortskurve des Frequenzganges des aufgeschnittenen, linearen Regelkreises und der Voraussetzungen über die Charakteristik des nichtlinearen Gliedes zurückgeführt wird. Auf diese Weise wird ein Zusammenhang mit den bekannten Frequenzstabilitätskriterien selbsttätiger Regelungssysteme hergestellt, die ein nichtlineares Element erhalten.
    ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 11/2006; 49(3):167 - 177. · 0.95 Impact Factor
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    ABSTRACT: Algorithms are developed which compute the stability boundaries of the equilibrium points of biotechnological reaction systems and which visualize these boundaries in a high dimensional state space. The algorithms predict the ultimate process behaviour from its present state and from the control input levels, using an analytical model of the system. In particular, they allow to forecast whether the process will converge to a nominal operational state or to a condition of biological wash out. The paper concentrates on systems of rank two, which encompass a wide range of potential industrial applications.
    06/2006;
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    ABSTRACT: The dynamics of biochemical reaction systems of rank one are studied. The global convergence of the set of equilibrium points is investigated and the local stability properties of the equilibria are analysed. The problem of stability boundary estimation is addressed and an algorithm for the visualization of the boundaries is proposed. The theoretical results and the effectiveness of the proposed algorithms are verified by two examples.
    Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on; 01/2006
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    ABSTRACT: This paper addresses the dynamics of a SHARON bioreactor for ammonium removal from concentrated wastewater streams. It is shown that multiple equilibrium points occur for a simplified reactor model. Conditions are determined for which the system possesses multiple equilibrium points and the corresponding phase portraits are analysed. In case the reactor model possesses two locally asymptotically stable equilibrium points, the stability boundary, that separates their attraction regions, is visualized. Subsequently, it is examined how small parameter changes affect the number of equilibrium points and the corresponding phase portraits. The analytically obtained results are illustrated by means of simulations.
    Nonlinear Dynamics and Systems Theory 01/2006; 6(2):191-204.
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    ABSTRACT: This paper addresses the dynamics of a SHARON reactor, a promising technology for ammonium removal from concentrated wastewater streams. The contraction mapping theorem is used to determine which operating conditions of a SHARON reactor with pH-control result in a unique equilibrium state. However, this approach only identifies the case of very large dilution rates, in practice corresponding with complete biomass wash-out, i.e. with complete loss of biological activity. Practical operation of a SHARON reactor aims at reaching ammonium conversion to nitrite. To identify such interesting operating points, the equilibrium points are subsequently calculated directly in terms of input variables for a simplified SHARON reactor model. The stability of the obtained equilibrium points is assessed and the corresponding phase portraits are analyzed. The influence of slightly varying parameter and input values is investigated as well.
    Journal of Process Control 01/2006; 16(10):1003-1012. · 2.18 Impact Factor
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    L. Luyckx, M. Loccufier, E. Noldus
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    ABSTRACT: A numerical method is presented for the estimation and the visualization of the stability boundary of an equilibrium point in the state space of autonomous nonlinear dynamical systems. The technique uses a combination of certain concepts from Liapunov theory, numerical simulation and the properties of the geometric structure of the stability boundary. The boundary can be visualized by computing its intersections with arbitrarily selected two-dimensional planes in state space. The visualization problem is reduced to a two-point boundary value differential problem which is solved using a flooding technique.
    Journal of Computational and Applied Mathematics 07/2004; 168(s 1–2):289–297. · 0.99 Impact Factor
  • L. Luyckx, M. Loccufier, E. Noldus
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    ABSTRACT: We are interested in the output feedback control of mechanical systems governed by the Euler-Lagrange formalism. The systems are collocated actuator-sensor controlled and underactuated. We present a design method by means of a specific example : the set point control of a rotating pendulum. We use constrained output feedback, whereby the control inputs satisfy a priori imposed upper bounds. The closed loop stability analysis relies on the direct method of Liapunov. This results in a frequency criterion on the controller's linear dynamic component and some restrictions on its nonlinearities. The control parameters are tuned for maximizing closed loop damping.
    01/2004;
  • Proceedings of the 22nd IASTED International Conference on Modelling, Identification, and Control (MIC 2003), February 10-13, 2003, Innsbruck, Austria; 01/2003
  • M. Loccufier, E. Noldus
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    ABSTRACT: A new systems theoretical approach to the design of passive vibration absorbers for elastic systems is presented. Undamped lumped parameter models of flexible mechanical structures are considered. Necessary and sufficient conditions are derived under which a preselected eigenfrequency spectrum can be assigned to such a system using an undamped lumped parameter vibration absorber. The realizable eigenfrequency spectra are analytically characterized, the required dynamical order of the absorber is established, and a design method for its parameters is outlined.
    ZAMM Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 01/2001; 81(8):541-547. · 0.95 Impact Factor
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    ABSTRACT: The suppression of oscillations in a class of generalized gradient systems using nonlinear dynamic output feedback is investigated. A class of controllers is considered which, in addition to a linear dynamic component, possess several types of nondynamic nonlinearities. Frequency domain conditions on the transfer matrix of the controller's linear component are presented that ensure the convergence of all closed-loop solutions to an equilibrium point in state space, thus eliminating the occurrence of sustained oscillations. Practically important technical applications include a.o. the set point control of mechanical systems described by the Euler–Lagrange equations and their equivalent Hamiltonian formulation. The obtained results constitute a systems theoretical basis for a new method of nonlinear vibration controller design.
    Cybernetics and Systems 01/2001; 32:811-827. · 0.97 Impact Factor
  • M. Loccufier, E. Noldus
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    ABSTRACT: A new method is presented for estimating regions of asymptoticstability for autonomous nonlinear dynamical systems. The underlyinganalysis uses a combination of Lyapunov theory, simulation and sometopological properties of the stability boundary. The advantages of themethod are the accuracy of estimation of the true stability boundary,its numerical robustness and its applicability to wide classes ofdynamical systems. The main limitation is that a global Lyapunovfunction for the system must be available.
    Nonlinear Dynamics 02/2000; 21(3):265-288. · 3.01 Impact Factor