Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept

The purpose of the paper is to generalize the results obtained by the authors in their last works which are related to the asymptotic properties of the pseudoinverse model-based method for designing an efficient steady-state control of interconnected systems with uncertainties and arbitrary bounded disturbances and also to present some new results.Розглянуто концепцію псевдообернення як деяку уніфіковану концепцію керування усталеними станами багатозв'язних систем за наявності невимірюваних обмежених збурень з повною і неповною інформацією про параметри лінійної номінальної моделі, по якій будується зворотний зв'язок. Припускається, що ранг матриці коефіцієнтів підсилення цієї моделі може бути довільним. Встановлено достатні умови граничної обмеженості всіх сигналів у замкнених системах керування, що реалізують запропоновану концепцію. Наведено результати моделюванняРассмотрена концепция псевдообращения как некоторая унифицированная концепция управления установившимися состояниями многосвязных систем при наличии неизмеряемых ограниченных возмущений с полной и неполной информацией о параметрах линейной номинальной модели, по которой строится обратная связь. Предполагается, что ранг матрицы коэффициентов усиления этой модели может быть произвольным Установлены достаточные условия предельной ограниченности всех сигналов в замкнутых системах управления, реализующих предлагаемую концепцию. Приведены результаты моделирования

Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank

The purpose of this paper is to answer the question of how the pseudoinverse modelbased adaptive approach might be utilized to deal with the uncertain multivariable memoryless system if the number of control inputs is less than the number of outputs. Results. It is shown that the parameter estimates generated by the standard adaptive projection recursive procedure converge always to some finite values for any initial values of system’s parameters. Based on these ultimate features, it is proved that the adaptive pseudoinverse model-based control law makes it possible to achieve the equilibrium state of the nonsquare system to be controlled. The asymptotical properties of the adaptive feedback control system derived theoretically are substantiated by a simulation experiment.Метою даного дослідження є відповідь на питання про те, чи можна реалізувати адаптивний підхід на основі псевдооберненої моделі для керування невизначеною багатомірною системою без пам'яті, в якій кількість входів керування є менша за кількість вихідних змінних. Результати. Показано, що оцінки параметрів, які формуються стандартною адаптивною рекурентною процедурою проекційного типу, завжди збігаються до деяких скінченних значень за будь-яких початкових оцінок параметрів системи. Доведено, що адаптивний закон керування на основі псевдооберненої моделі дозволяє досягти положення рівноваги системи, яка підлягає керуванню. Асимптотичні властивості системи керування з адаптивним зворотним зв'язком, встановлені теоретично, підтверджуються модельним експериментом.Цель этой статьи — ответить на вопрос, можно ли реализовать адаптивный подход на основе псевдообратной модели для управления неопределенной многомерной системой без памяти, в которой число управляющих входов меньше числа выходных переменных. Результаты. Показано, что оценки параметров, генерируемые стандартной адаптивной рекуррентной процедурой проекционного типа, всегда сходятся к некоторым конечным значениям для любых начальных оценок параметров системы. Доказано, что адаптивный псевдообратный закон управления позволяет достичь положения равновесия управляемой системы. Асимптотические свойства адаптивной системы управления с обратной связью, полученные теоретически, подтверждены модельным экспериментом

Maximum-likelihood estimation of lithospheric flexural rigidity, initial-loading fraction, and load correlation, under isotropy

Topography and gravity are geophysical fields whose joint statistical structure derives from interface-loading processes modulated by the underlying mechanics of isostatic and flexural compensation in the shallow lithosphere. Under this dual statistical-mechanistic viewpoint an estimation problem can be formulated where the knowns are topography and gravity and the principal unknown the elastic flexural rigidity of the lithosphere. In the guise of an equivalent "effective elastic thickness", this important, geographically varying, structural parameter has been the subject of many interpretative studies, but precisely how well it is known or how best it can be found from the data, abundant nonetheless, has remained contentious and unresolved throughout the last few decades of dedicated study. The popular methods whereby admittance or coherence, both spectral measures of the relation between gravity and topography, are inverted for the flexural rigidity, have revealed themselves to have insufficient power to independently constrain both it and the additional unknown initial-loading fraction and load-correlation fac- tors, respectively. Solving this extremely ill-posed inversion problem leads to non-uniqueness and is further complicated by practical considerations such as the choice of regularizing data tapers to render the analysis sufficiently selective both in the spatial and spectral domains. Here, we rewrite the problem in a form amenable to maximum-likelihood estimation theory, which we show yields unbiased, minimum-variance estimates of flexural rigidity, initial-loading frac- tion and load correlation, each of those separably resolved with little a posteriori correlation between their estimates. We are also able to separately characterize the isotropic spectral shape of the initial loading processes.Comment: 41 pages, 13 figures, accepted for publication by Geophysical Journal Internationa

Adjoint-state method for seismic AVO inversion and time-lapse monitoring

This dissertation presents seismic amplitude versus offset (AVO) inversion methods to estimate water saturation and effective pressure quantitatively in elastic and viscoelastic media. Quantitative knowledge of the saturation and pore pressure properties from pre- or post-production seismic measurements for reservoir static or dynamic modeling has been an area of interest for the geophysical community for decades. However, the focus on the existing inversion methodologies and explicit expressions to estimate saturation-pressure variables or changes in these properties due to production or fluid injection has been based on elastic AVO models. These conventional methods do not consider the seismic wave attenuation effects on the reflection amplitudes and therefore can result in biased prediction. Numerous theoretical rock physics models and laboratory experiments have demonstrated the sensitivity of various petrophysical and seismic properties of partially fluid-filled porous media to seismic attenuation. This makes seismic wave attenuation a valuable time-lapse attribute to reliably measure the saturation (Sw) and effective pressure (Pe) properties. Therefore, in this work, I have developed two AVO inversion processes i.e., the conventional AVO inversion method for elastic media and the frequency-dependent amplitude versus offset (FAVO) inversion technique for the viscoelastic media. This dissertation first presents the inversion strategies to invert the pre-stack seismic data for the seismic velocities and density by using the conventional AVO equation and for the seismic velocities, density, and Q-factors by using the frequency-dependent AVO method. These inversion methods are then extended to estimate the dynamic reservoir changes e.g., saturation and pressure variables, and can be applied to predict the saturation and pressure variables at any stage e.g., before and during production, or fluid injection, or to estimate the changes in saturation (ΔSw) and pressure (ΔPe). The first part of the dissertation describes the theory and formulation of the elastic AVO inversion method while in the second half, I have described the viscoelastic inversion workflow. FAVO technique accounts for the dependence of reflection amplitudes on incident angles as well as seismic frequencies and P and S waves attenuation in addition to seismic velocities and density. The fluid saturation and pressure in the elastic and inelastic mediums are linked to the reflection amplitude through seismic velocities, density, and quality factors (Q). The inversion process is based on the gradient-descent method in which the least-square differentiable data misfit equation is minimized by using a non-linear limitedmemory BFGS method. The gradients of the misfit function with respect to unknown model variables are derived by using the adjoint-state method and the multivariable chain rule of derivative. The adjoint-state method provides an efficient and accurate way to calculate the misfit gradients. Numerous rock physics models e.g., the Gassmann substitution equation with uniform and patchy fluid distribution patterns, modified MacBeth’s relations of dry rock moduli with effective pressure, and constant Q models for the P and S wave attenuation are applied to relate the saturation and effective pressure variables with elastic and an-elastic properties and then forward reflectivity operator. These inversion methods have been defined as constrained problems wherein the constraints are applied e.g., bound constraints, constraints in the Lagrangian solution, and Tikhonov regularization. These inversion methods are quite general and can be extended for other rock physics models through parameterizations. The applications of the elastic AVO and the FAVO methods are tested on various 1D synthetic datasets simulated under different oil production (4D) scenarios. The inversion methods are further applied to a 2D realistic reservoir model extracted from the 3D Smeaheia Field, a potential storage site for the CO2 injection. The inversion schemes successfully estimate not only the static saturation and effective pressure variables or changes in these properties due to oil production or CO2 injection but also provide a very good prediction of seismic velocities, density, and seismic attenuation (quantified as the inverse quality factor). The partially CO2-saturated reservoir exhibits higher P wave attenuation, therefore, the addition of time-lapse P wave attenuation due to viscous friction between CO2-water patches helps to reduce the errors in the inverted CO2/water saturation variables as compared to the elastic 4D AVO inversion. This research work has a wide range of applications from the oil industry to carbon capture and storage (CCS) monitoring tools aiming to provide control and safety during the injection. The uncertainty in the inversion results is quantified as a function of the variability of the prior models obtained by using Monte Carlo simulation