Pilot in-the-loop oscillation (PIO) is a phenomenon which occurs due to the dynamic interaction between pilot and aircraft. This detrimental aircraft handling quality appears through a variety of flight conditions and is very difficult to predict. Due to this complex behavior, PIO is not easily eliminated. This report describes a method of PIO analysis that is capable of examining multiple asymmetric nonlinearities acting simultaneously. PIO analyses are performed on a model based on the USAF NT-33A variable stability aircraft with nonlinearities including stick position limiting, elevator deflection limiting, and elevator rate limiting. These analyses involve the use of dual input describing functions which enable the prediction of frequency, amplitude, and mean point of oscillation. / Master of Science

No description available.

In this thesis, low-order nonlinear models for the prediction of the nonlinear behaviour of thermoacoustic systems are developed. These models are based on thermoacoustic networks, in which linear acoustics is combined with a nonlinear heat release model. The acoustic networks considered in this thesis can take into account mean flow and non-trivial acoustic reflection coefficients, and are cast in state-space form to enable analysis both in the frequency and time domains. Starting from linear analysis, the stability of thermoacoustic networks is investigated, and the use of adjoint methods for understanding the role of the system's parameters on its stability is demonstrated. Then, a nonlinear analysis using various state-of-the-art methods is performed, to highlight the strengths and weaknesses of each method. Two novel frameworks that fill some gaps in the available methods are developed: the first, called Flame Double Input Describing Function (FDIDF), is an extension of the Flame Describing Function (FDF). The FDIDF approximates the flame nonlinear response when it is forced simultaneously with two frequencies, whereas the FDF is limited to one frequency. Although more expensive to obtain, the FDIDF contains more nonlinear information than the FDF, and can predict periodic and quasiperiodic oscillations. It is shown how, in some cases, it corrects the prediction of the FDF about the stability of thermoacoustic oscillations. The second framework developed is a weakly nonlinear formulation of the thermoacoustic equations in the Rijke tube, in which the acoustic response is not limited to a single-Galerkin mode, and is embedded in a state-space model. The weakly nonlinear analysis is strictly valid only close to the expansion point, but is much cheaper than any other available method. The above methods are applied to relatively simple thermoacoustic configurations, in which the nonlinear heat release model is that of a laminar conical flame or an electrical heater. However, in real gas turbines more complex flame shapes are found, for which no reliable low-order models exist. Two models are developed in this thesis for turbulent bluff-body stabilised flames: one for a perfectly premixed flame, in which the modelling is focused on the flame-flow interaction, accounting for the presence of recirculation zones and temperature gradients; the second for imperfectly premixed flames, in which equivalence ratio fluctuations, modelled as a passive scalar field, dominate the heat release response. The second model was shown to agree reasonably well with experimental data, and was applied in an industrial modelling project.

Lean-premixed combustion has the advantage of low emissions for modern gas turbines, but it is susceptible to thermoacoustic instabilities, which can result in large amplitude pressure oscillations in the combustion chamber. The thermoacoustic limit cycle is generated by the unsteady heat release dynamics coupled to the combustor acoustics. In this dissertation, we focused on reduced-order modeling of the dynamics of a laminar premixed flame. From first principles of combustion dynamics, a physically-based, reduced-order, nonlinear model was developed based on the proper orthogonal decomposition technique and generalized Galerkin method. In addition, the describing function for the flame was measured experimentally and used to identify an empirical nonlinear flame model. Furthermore, a linear acoustic model was developed and identified for the Rijke tube experiment. Closed-loop thermoacoustic modeling using the first principles flame model coupled to the linear acoustics successfully reproduced the linear instability and predicted the thermoacoustic limit cycle amplitude. With the measured experimental flame data and the modeled linear acoustics, the describing function technique was applied for limit cycle analysis. The thermoacoustic limit cycle amplitude was predicted with reasonable accuracy, and the closed-loop model also predicted the performance for a phase shift controller. Some problems found in the predictions for high heat release cases were documented. / Ph. D.

This thesis presents the fundamental issues of the digital controlled DC/DC converter. A lot of challenges exist when you introduce the digital control technique into the control of the DC/DC converter, especially with regards to the voltage regulator module. One issue is the limit cycle oscillation problem caused by the quantization effect from the ADC and DPWM of the digital control chip. Another issue is the delay problem coming from the sample-hold effect. In this thesis, the modeling, analysis and design methodology for the constant frequency voltage-mode control is reviewed. A DPWM (Digital Pulse Width Modulator) model is verified in simulation, which shows what effects the digital control brings to the conventional Pulse Width Modulator. In CPES, the constant on-time control concept is introduced into the digital control of the voltage regulator module. This provides a high resolution of DPWM and allows the digital constant on-time voltage-mode control architecture to be proposed. To limit the oscillation amplitude in the digital control structure, the digital constant on-time current-mode control w/ external ramp is further proposed in CPES. To analyze this structure, a describing function model is proposed for the digital constant on-time current-mode control, which takes both the sample-hold effect and the quantization effect into consideration. This model clearly shows the stability problem caused by the sample-hold effect in the current loop. Using larger ramp's slope values, this stability issue can be alleviated. Based on this model, a design methodology is introduced. By properly designing the current loop's ADC resolution and the voltage loop's ADC resolution, the limit cycle oscillation in this structure can be minimized: the digital constant on-time current-mode control will only have the oscillation coming from the sample-hold effect in the current loop, which can be greatly reduced by adding the large slope's external ramp to this structure. Simulation verification for this design methodology is provided to prove the concepts. Based on the proposed model, the compensator design is performed. The motivation for the compensator design is to push the bandwidth while satisfying the stability condition and the dynamic no-limit-cycle oscillation condition. When analyzing the case of one sample per switching cycle, there is a certain amount of delay, which compromises the phase characteristics. Our design also requires a large external ramp because it will reduce the oscillation amplitude in our system. From our model, it is quite obvious that the external ramp must have a slope larger than one time that of inductor current down slope. A slope that is too larger will weaker the phase and limit the bandwidth. When using the normal current-mode compensator, like the 1-pole 1-zero compensator, the phase is dropped too much and the bandwidth will be limited too low. If we use a 2-pole 2-zero compensator, the phase can be boosted. However, in this case, the gain margin requirement from the dynamic no-limit-cycle oscillation condition will make the further improvements on bandwidth impossible. In our design, the one sixth of the switching frequency is achieved. / Master of Science

Cette thèse se concentre sur l’étude des instabilités de combustion dans un brûleur prémélangé. Les instabilités sont généralement issues d’un couplage entre la combustion et les modes propres du système. La mise en résonance qui en résulte peut avoir des conséquences qui sont souvent dommageables, entraînant des vibrations, une fatigue des matériaux soumis à des charges acoustiques élevées et une intensification des flux de chaleur vers les parois de la chambre. Un premier objectif de cette thèse est de poursuivre le développement de méthodes de prévision des instabilités et des phénomènes non-linéaires qui en résultent comme par exemple le développement de cycles limites, les processus de déclenchement (“triggering”), la commutation de modes. Le cadre général adopté est celui de «°l’équivalent harmonique » bien connu dans le domaine du contrôle et qui a été exploré dans le domaine des instabilités de combustion dans des travaux récents du laboratoire EM2C, CNRS. Par le biais de ce concept il est possible de tenir compte de l’´evolution de la réponse de la flamme suivant l’amplitude à laquelle elle est soumise. Cette réponse de flamme en fréquence et amplitude généralise la notion de fonction de transfert et elle est désignée sous le nom de “Flame Describing Function” (FDF). Le système est ouvert à son extrémité aval. Cette géométrie permet de simplifier l’analyse et d’obtenir une large gamme de configurations au moyen d’une variation continue de la longueur du conduit d’alimentation qui est limité en amont par un piston. On peut aussi échanger le tube à flamme et utiliser des longueurs différentes de cet élément. Une étude exhaustive est réalisée pour répertorier les oscillations observées et déduire leurs propriétés. On montre que les cycles limites qui possèdent une amplitude constante sont bien décrits par la méthode unifiée fondée sur la FDF. Pour certaines configurations l’expérience fait apparaître des cycles limites dont l’amplitude et la fréquence ne se stabilisent pas au cours du temps. On observe notamment des oscillations plus complexes couplées par plusieurs modes pouvant soit donner lieu à des variations régulières ou à des fluctuations plus irrégulières avec un caractère “galopant” dans le temps. Pour ces oscillations particulières, la FDF fournit des indications sur les domaines d’apparition mais n’est pas en mesure de décrire complètement ces cycles limites complexes. Il faut dans ce cas recourir à une représentation temporelle qui n’est pas développée dans ce document. La base de données expérimentales pourra être utilisée pour guider ultérieurement ce type d’analyse. Le deuxième grand objectif de cette thèse est de rechercher des méthodes de contrôle des instabilités. On considère plus particulièrement des systèmes dynamiques utilisant des plaques perforées polarisées par un écoulement (BFP : “bias flow perforate”). Ces systèmes sont particulièrement intéressants pour atténuer les oscillations basse fréquence qui sont difficiles à réduire par des systèmes passifs. La conception de ces BFPs est fondée sur des travaux récents menés au laboratoire EM2C, CNRS avec notamment l’objectif de robustesse, c’est-à-dire la possibilité de couvrir une large bande de fréquences. L’´etude expérimentale et les calculs fondés sur la FDF menés en parallèle permettent de voir les possibilités de tels systèmes et de comprendre les conditions nécessaires à leur efficacité. Cette étude peut permettre de guider les applications qui pourraient être envisagées en pratique. / This thesis is concerned with an investigation of combustion instabilities in premixed combustors. This problem has been the subject of a continuous effort in relation with the many issues encountered in practical systems like those used in propulsion and energy production. Combustion instabilities usually arise from the coupling between combustion and acoustic eigenmodes of the system. In most cases such resonances lead to vibrations, structural fatigue and intensified heat fluxes to the chamber walls. The first part of this thesis pursues the development of prediction methods for combustion instabilities and the associated nonlinear phenomena such as limit cycles establishment, triggering, mode switching and hysteresis. The aim is to delineate physical mechanisms and develop analytical methods dedicated to prediction. The theoretical framework relies on the “harmonic balance” formalism well known in the domain of control and which has been adopted more recently in combustion instability studies carried out at EM2C, CNRS laboratory. Through this concept, it is possible to take into account the evolution of the flame response as a function of amplitude. This flame response, depending on frequency and amplitude, extends the flame transfer function principle and is designated as the “Flame Describing Function” (FDF). The development of the FDF framework is pursued in the present study. The experimental setup which exemplifies combustion instabilities and serves to validate the method has generic features as it comprises in an idealized version, all the parts found in practical systems : a feeding manifold delivering a mixture of methane and air, a multipoint injector made of a perforated plate anchoring a collection of small laminar conical flames and a flame tube made of quartz which confines the combustion zone. The downstream boundary of the system is open. This device allows a simplified analysis and provides a wide variety of configurations through the continuous modification of the feeding manifold length which is bounded by a piston on the upstream and through changes of the flame tube lengths. Systematic comparison between theoretical results and well controlled experiments is undertaken. Depending on the geometry, the setup exhibits a large variety of unstable modes which are classified in terms of their limit cycle behavior using tools from dynamical system theory. It is shown that limit cycles with constant amplitude are well predicted by the unified FDF methodology. For some configurations, the experiment reveals limit cycles characterized by time variable amplitude and frequency. One finds situations where the oscillation is coupled by multiple modes leading either to regular amplitude variations or more irregular evolutions with a “galloping” pattern as a function of time. For this special type of limit cycle, the FDF indicates the range of the onset, but is not able to fully describe these complex limit cycles. These oscillations require a time domain state space analysis which is not addressed in this manuscript. The experimental database may be of value for further work in this direction. The second part of this thesis deals with control methods for instabilities. One specifically considers damping systems relying on perforated plates biased by a flow (BFP : “Bias Flow Perforate”). These systems are particularly interesting because they can be used to cancel low frequency oscillations which are otherwise difficult to reduce through passive control methods. This BFP design relies on recent work carried out at EM2C, CNRS laboratory which extends the frequency range where the system is effective. The experimental study and the associated FDF calculations are used to delineate the possibilities of such systems and uncover conditions required for an effective damping of oscillations. This study provides indications on the practical application of BFPs.

No description available.

Due to unique characteristics, current-mode control architectures with different implementation approaches have been widely used in power converter design to achieve current sharing, AVP control, and light-load efficiency improvement. Therefore, an accurate model for current-mode control is indispensable to system design due to the existence of subharmonic oscillations. The fundamental difference between current-mode control and voltage-mode control is the PWM modulation. The inductor current, one of state variables, is used in the modulator in current-mode control while an external ramp is used in voltage-mode control. The dynamic nonlinearity of current-mode control results in the difficulty of obtaining the small-signal model for current-mode control in the frequency domain. There has been a long history of the current-mode control modeling. Many previous attempts have been made especially for constant-frequency peak current-mode control. However, few models are available for variable-frequency constant on-time control and V2 current-mode control. It's hard to directly extend the model of peak current-mode control to those controls. Furthermore, there is no simple way of modeling the effects of the capacitor ripple which may result in subharmonic oscillations in V2 current-mode control. In this dissertation, the primary objective to investigate a new and general modeling approach for current-mode control with different implementation methods. First, the fundamental limitation of average models for current-mode control is identified. The sideband components are generated and coupled with the fundamental component through the PWM modulator in the current loop. Moreover, the switching frequency harmonics cannot be ignored in the current loop since the current ripple is used for the PWM modulation. Available average models failed to consider the sideband effects and high frequency harmonics. Due to the complexity of the current loop, it is difficult to analyze current loop in the frequency domain. A new modeling approach for current-mode control is proposed based on the time-domain analysis. The inductor, the switches and the PWM modulator are treated as a single entity to model instead of breaking them into parts to do it. Describing function method is used. Proposed approach can be applied not only to constant-frequency modulation but also to variable-frequency modulation. The fundamental difference between different current-mode controls is elaborated based on the models obtained from the new modeling approach. Then, an equivalent circuit representation of current-mode control is presented for the sake of easy understanding. The effect of the current loop is equivalent to controlling the inductor current as a current source with certain impedance. The circuit representation provides both the simplicity of the circuit model and the accuracy of the proposed model. Next, the new modeling approach is extended to V2 current-mode control based on similar concept. The model for V2 current-mode control can accurately predict subharmonic oscillations due to the influence of the capacitor ripple. Two solutions are discussed to solve the instability issue. After that, a digital application of current-mode control is introduced. High-resolution digital pulse-width modulator (DPWM) is considered to be indispensable for minimizing the possibility of unpredicted limit-cycle oscillations, but results in high cost, especially in the application of voltage regulators for microprocessors. In order to solve this issue, a fully digital current-mode control architecture which can effectively limit the oscillation amplitude is presented, thereby greatly reducing the design challenge for digital controllers by eliminating the need for the high-resolution DPWM. The new modeling strategy is also used to model the proposed digital current-mode control to help system design. As a conclusion, a new modeling approach for current-mode control is fully investigated. Describing function method is utilized as a tool in this dissertation. Proposed approach is quite general and not limit by implementation methods. All the modeling results are verified through simulation and experiments. / Ph. D.

Various types of current mode control are being used in different applications. Model for current mode control is indispensable for proper system design. Since 1980s, modeling of current mode control has been a hot topic in power electronics field. In current mode control, sub-harmonic oscillation is a common issue, especially for constant frequency current mode control: like peak current mode control, valley current mode control, or average current mode control. Recently V2 control is becoming more and more popular due to its simple implementation ad super fast transient response. V2 control can also run into sub-harmonic oscillation just as current mode control. Efforts have been devoted to modeling of V2 control. A common property of different types of current mode control and V2 control is that they are all multi-loop structures and the inner loops are all highly nonlinear. Due to the nonlinearity of the inner loops, modeling of these structures is extremely difficult. Up to now, there are two main problems which haven't been solved: 1. modeling of average current mode control; 2. modeling of V2 control with composite capacitors. This thesis tries to solve these two problems and starts with V2 control. For V2 control with single type of bulk capacitors, an accurate model has been proposed previously. In this thesis, an equivalent circuit model is proposed to get better physical understanding. This method makes use of previous current mode control modeling result and relates V2 control with current mode control. To model V2 control with composite capacitors, capacitor currents and output voltage time domain waveforms are analyzed. Based on describing function method, transfer function from control to output is derived. The modeling result shows that with more parallel ceramic capacitors, system has smaller stability margin. For average current mode control, the structure is compared with V2 control. Similarity between the structures of current compensator in average current mode and output capacitor network in V2 control is identified. V2 model is utilized for average current mode control. The modeling derivation process is simplified. For the current compensator in average current mode control, it is not desired to have a high frequency pole from stability point of view. As a conclusion, a circuit model for V2 control with bulk capacitors is proposed and another two problems are examined: modeling of V2 control with composite capacitors and modeling of average current mode control. It has been demonstrated that there is similarity between these two structures. The modeling results are verified through simulation and experiments. / Master of Science

<p>Data networks are an important part in an increasing number of applications with real-time and reliability requirements. To meet these demands a variety of approaches have been proposed. Forward error correction, which adds redundancy to the communicated data, is one of them. However, the redundancy occupies communication bandwidth, so it is desirable to control the amount of redundancy in order to achieve high reliability without adding excessive communication delay. The main contribution of the thesis is to formulate the problem of adjusting the redundancy in a control framework, which enables the dynamic properties of error correction control to be analyzed using control theory. The trade-off between application quality and resource usage is captured by introducing an optimal control problem. Its dependence on the knowledge of the network state at the transmission side is discussed. An error correction controller that optimizes the amount of redundancy without relying on network state information is presented. This is achieved by utilizing an extremum seeking control algorithm to optimize the cost function. Models with varying complexity of the resulting feedback system are presented and analyzed. Conditions for convergence are given. Multiple-input describing function analysis is used to examine periodic solutions. The results are illustrated through computer simulations and experiments on a wireless sensor network.</p>