Mia Loccufier

A new trajectory reversing method for the estimation of asymptotic stability regions

A new method is presented for estimating regions of asymptotic stability for autonomous nonlinear systems. The method applies to systems that can be proven to be non-oscillatory, using a suitable Lyapunov technique. It is especially effective for second-order systems that possess several equilibrium states for which the regions of attraction must be found. The determination of the regions of asymptotic stability requires the explicit computation of a Lyapunov function and the determination by backward numerical integration of a limited number of trajectories. The obtained estimates are close approximations of the exact regions of asymptotic stability of the systems stable equilibria.

A New Trajectory Reversing Method for Estimating Stability Regions of Autonomous Nonlinear Systems

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.

An optimizing start-up strategy for a bio-methanator

This paper presents an optimizing start-up strategy for a bio-methanator. The goal of the control strategy is to maximize the outflow rate of methane in anaerobic digestion processes, which can be described by a two-population model. The methodology relies on a thorough analysis of the system dynamics and involves the solution of two optimization problems: steady-state optimization for determining the optimal operating point and transient optimization. The latter is a classical optimal control problem, which can be solved using the maximum principle of Pontryagin. The proposed control law is of the bang–bang type. The process is driven from an initial state to a small neighborhood of the optimal steady state by switching the manipulated variable (dilution rate) from the minimum to the maximum value at a certain time instant. Then the dilution rate is set to the optimal value and the system settles down in the optimal steady state. This control law ensures the convergence of the system to the optimal steady state and substantially increases its stability region. The region of attraction of the steady state corresponding to maximum production of methane is considerably enlarged. In some cases, which are related to the possibility of selecting the minimum dilution rate below a certain level, the stability region of the optimal steady state equals the interior of the state space. Aside its efficiency, which is evaluated not only in terms of biogas production but also from the perspective of treatment of the organic load, 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 anaerobic digestion process, for which the acidogenesis and methanogenesis are, respectively, characterized by Monod and Haldane kinetics.

On the Optimization of Biogas Production in Anaerobic Digestion Systems

Abstract This paper presents a strategy for the optimization of biogas outflow rate in an anaerobic digestion process described by a two-population model. The methodology relies on the solution of two optimization problems: steady state optimization for determining the optimal operating point and transient optimization. The latter is solved using the maximum principle of Pontryagin. The proposed control law, which drives the process from an initial state to the optimal steady state while maximizing the biogas outflow rate, 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, enlarging it in some cases to almost the entire state space. 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 anaerobic digestion process, for which the acidogenesis and methanogenesis are respectively characterized by Monod and Haldane kinetics.

Sensor design for outdoor racing bicycle field testing for human vibration comfort evaluation

This paper is concerned with the vibrational comfort evaluation of the cyclist when cycling a rough surface. Outdoor comfort tests have so far only been done through instrumenting the bicycle with accelerometers. This work instruments a racing bicycle with custom-made contact force sensors and velocity sensors to acquire human comfort through the absorbed power method. Comfort evaluation is assessed at the hand–arm and seat interface of the cyclist with the bicycle. By means of careful finite-element analysis for designing the force gauges at the handlebar and the seat combined with precise calibration of both force and velocity sensors, all sensors have proven to work properly. Initial field tests are focused on the proper functioning of the designed sensors and their suitability for vibration comfort measurements. Tests on a cobblestone road reveal that the outcome of the absorbed power values is within the same range as those from laboratory tests found in the literature. This sensor design approach for outdoor testing with racing bicycles may give a new interpretation on evaluating the cyclists comfort since the vibrational load is not only quantified in terms of acceleration but also in terms of force and velocity at the bicycle–cyclist contact points.

Steady state multiplicity of two-step biological conversion systems with general kinetics

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 models 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.

Stability of analogue neural classification networks

A general frequency domain stability criterion is presented for autonomous nonlinear dynamical systems that possess several non-linearities and several equilibrium states. As special cases, the result is shown to contain well-known criteria for non-oscillatory behaviour of non-linear feedback loops, of non-linear electrical RLC-circuits and analogue neural networks. The proof relies on a single Liapunov function which can subsequently be used lo compute regions of attraction for each of the equilibrium states. Therefore the result is particularly useful for the analysis and the design of systems such as neural classification networks, which possess many equilibrium states.

The Estimation of Stability Boundaries for an Anaerobic Digestion System

Abstract The operation of biochemical reaction systems requires substantial expertise and a good understanding of system dynamics. Due to various inhibition effects, these systems usually possess several equilibrium points. Selecting the initial state of the system is as important as selecting the systems 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 systems inputs. The systems 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 acidification 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 systems operator.

On the Attachment Location of Dynamic Vibration Absorbers

In mechanical engineering a commonly used approach to attenuate vibration amplitudes in resonant conditions is the attachment of a dynamic vibration absorber. The optimal parameters for this damped spring-mass system are well known for single degree of freedom undamped main systems (Den Hartog). An important parameter when designing absorbers for multi degree of freedom systems is the location of the absorber, i.e. where to physically attach it. This parameter has a large influence on the possible vibration reduction. Often however, anti nodal locations of a single mode are a priori taken as best attachment locations. This single mode approach loses accuracy when dealing with a large absorber mass or systems with closely spaced eigenfrequencies. To analyze the influence of the neighboring modes, the effect the absorber has on the eigenfrequencies of the undamped main system is studied. Given the absorber mass we determine the absorber locations that provide eigenfrequencies shifted as far as possible from the resonance frequency as this improves the vibration attenuation. It is shown that for increasing absorber mass, the new eigenfrequencies cannot shift further than the neighboring anti resonances due to interlacing properties. Since these anti resonances depend on the attachment location, an optimal location can be found. A procedure is described that yields the optimalabsorber location. This procedure combines information about the eigenvector of the mode to be controlled with knowledge about the neighboring anti resonances. As the neighboring anti resonances are a representation of the activity of the neighboring modes, the proposed method extends the commonly used single mode approach to a multi mode approach. It seems that in resonance, a high activity of the neighboring modes has a negative effect on the vibration reduction.

An application of Liapunov's method for the analysis of neural networks

The stability is studied of a class of nonlinear dynamical systems which possess many nonlinearities and many equilibrium states. As a special case, the analyzed class of systems includes analog neural networks. Sufficient conditions for the nonoscillatory behaviour of these systems, in the form of frequency domain criteria, are presented. The main result is proved relying on a suitable Liapunov function which is subsequently used for the simultaneous computation of regions of attraction for each stable equilibrium.