Stable Inversion Research Articles

Precise Tracking for Continuous‐Time Non‐Minimum Phase Systems

AbstractStable inversion based precise tracking for continuous‐time square or nonsquare non‐minimum phase systems is studied. However, high precision trajectory tracking of non‐minimum phase systems can be obtained by the stable inversion method but requiring large enough extended time interval. In order to solve this problem of large extended time restriction, a novel approach to precise trajectory tracking of non‐minimum phase systems is proposed, it is called the improved stable inversion (ISI) method, using an optimal integration of the pre‐actuation and the optimal state transition (OST) techniques. The ISI method can obtain precise trajectory tracking with a smaller extended time interval as compared to the stable inversion method. The proposed method achieves better validation through numerical simulations for the non‐minimum phase system.

Organization and Oscillations in Simulated Shallow Convective Clouds

AbstractPhysical insights into processes governing temporal organization and evolution of cloud fields are of great importance for climate research. Here using large eddy simulations with a bin microphysics scheme, we show that warm convective cloud fields exhibit oscillations with two distinct periods (~10 and ~90 min, for the case studied here). The shorter period dominates the nonprecipitating phase, and the longer period is related to the precipitating phase. We show that rain processes affect the domain's thermodynamics, hence forcing the field into a low‐frequency recharge‐discharge cycle of developing cloudiness followed by precipitation‐driven depletion. The end result of precipitation is stabilization of the lower atmosphere by warming of the cloudy layer (due to latent heat release) and cooling of the subcloud layer (by rain evaporation, creating cold pools). As the thermodynamic instability weakens, so does the cloudiness, and the rain ceases. During the nonprecipitating phase of the cycle, surface fluxes destabilize the boundary layer until the next precipitation cycle. Under conditions that do not allow development of precipitation (e.g., high aerosol loading), high‐frequency oscillations dominate the cloud field. Clouds penetrating the stable inversion layer trigger gravity waves with a typical period of ~10 min. In return, the gravity waves modulate the clouds in the field by modifying the vertical velocity, temperature, and humidity fields. Subsequently, as the polluted nonprecipitating simulations evolve, the thermodynamic instability increases and the cloudy layer deepens until precipitation forms, shifting the oscillations from high to low frequency. The organization of cold pools and the spatial scale related to these oscillations are explored.

Reliable noise measure in time-gated NMR data

Summary Time gating is a commonly used approach in the pre-processing of nuclear magnetic resonance data before inversion. Gating suppresses spurious signals that can degrade recovered decay time distributions and therefore often stabilizes inversion. However, care must be taken in applying this technique when assumptions of uncorrelated Gaussianity break down and reduce stacking efficiency. If not properly accounted for, unreliable noise estimates introduce inversion artefacts through over- or underfitting of the data. Block-bootstrap resampling of noise realization proxies obtained through data phasing can be used to generate reliable noise estimates for the windows. Benefits of the approach are demonstrated through inversion of synthetic and borehole data. Analysis confirms that bootstrapped noise metrics are more reliable under variable noise conditions and result in more stable inversion results.

Precise tracking of periodic trajectories for periodic systems

Stable inversion was an effective method to achieve precise trajectory tracking, for which the tracking problem of periodically time-varying systems was solved in the existing literature. However, iteration techniques were required to approach stable inversion, whose explicit formulas for this tracking problem cannot be obtained. To overcome this drawback, a special kind of stable inversion, named periodic inversion, for the periodic systems is proposed in this study. By means of the lifted technique, the tracking problem of the periodic systems is reformulated to be equivalent to that of lifted systems. In order to obtain precise tracking, the periodic inversion of the periodic systems or the lifted systems is proposed, and is analysed for the non-singular case and the singular case, each of which is further analysed for three cases: the full row rank case, the full column rank case and the rank-deficient case. Thus the periodic inversion of the periodic systems is computed. Accompanied with the optimal state transition method for the time-varying systems, precise trajectory tracking is achieved. The methodology is validated through simulations.

On Two-Dimensional Hilbert Integral Equations, Generalized Minimum-Phase Signals, and Phase Retrieval

One-dimensional (1-D) causal signals admit Hilbert integral relations between the real and imaginary parts of their Fourier spectra. For 1-D minimum-phase signals, the log-magnitude and phase spectra also admit such Hilbert relations. In this paper, we extend these results to 2-D signals. We first establish the Hilbert integral equations for 2-D first-quadrant signals. For continuous-domain 2-D signals, we present the partial Hilbert transform relations between the real and imaginary parts of the spectrum. For the discrete-domain counterpart, we show that the partial Hilbert transform does not suffice, which motivates us to introduce the composite Hilbert transform . Considering the problem of phase retrieval, we show that, in both 1-D/2-D, and continuous/discrete domains, there exists a generalized class of signals, which subsumes the well-known class of 1-D minimum-phase signals with a rational transfer function, for which Hilbert integral relations exist between the log-magnitude and phase spectra. We introduce them under the nomenclature of generalized minimum-phase signals . For this class of signals, phase retrieval is possible by using the Hilbert transform without the requirement of a rational transfer function. Furthermore, we show that a generalized minimum-phase signal admits a stable convolutional inverse, which also belongs to the same class. Simulation results demonstrating accurate reconstruction of 1-D and 2-D generalized minimum-phase signals from their magnitude spectra are presented to support the theoretical developments.

An iterative method and its application to stable inversion

In this paper, we study convergence and data dependence of SP and normal-S iterative methods for the class of almost contraction mappings under some mild conditions. The validity of these theoretical results is confirmed with numerical examples. It has been observed that a special case of SP iterative method, namely normal-S iterative method, performs better and so the latter is implemented in the stable inversion of nonlinear discrete time dynamical systems to yield convergence results when Picard iterative method diverges. This is also illustrated with a numerical example. Our work extends and improves upon many results existing in the literature.

Synchronised trajectory tracking for a network of MIMO non‐minimum phase systems with application to aircraft control

In this study, the problem of synchronised trajectory tracking for a network of multi-input–multi-output (MIMO) non-minimum phase (NMP) systems is addressed under a switching communication topology. First, an approach to convert the original system equation into a normal form is proposed, where a coordinate transformation matrix is introduced. Then, a control solution is developed, which is made up of a distributed observer network to generate an estimate of the reference system, a causal stable inversion to estimate the bounded state reference and feedforward term, and a local state-trajectory tracker to achieve simultaneously asymptotic output tracking and internal dynamics stabilisation. Finally, the solution is applied to the synchronised flight-path angle and velocity tracking problem of a network of F-16 aircraft, and further compared with the classic proportional–integral–derivative control-based approach used in many real-life applications. Simulation results demonstrate the effectiveness of the proposed solution, as well as the performance advantages owing to a systematic consideration of the NMP property.

A behavioral approach to inversion-based control

Abstract A new simplified behavior theory is proposed to address inversion-based control for linear, nonminimum-phase SISO systems. The chosen space of signals is the set of piecewise C ∞ -functions and input–output pairs (as weak solutions) satisfy a differential–integral equation with additional smoothness requirements. A related key result is the output–input (or inverse) representation of the behavior set that leads to the solution of a general stable inversion problem where polynomially unbounded, noncausal desired outputs are allowed. It is shown that this problem has a solution if and only if the smoothness degree of the desired output is greater than or equal to the system relative degree minus one. When this straightforward condition is satisfied, a closed-form expression provides the inverse input. Then, an analysis on preaction and postaction control follows. Two examples are included showing the relevance of output signal design in control applications.

Three-dimensional Land FD-CSEM Regularized Inversion Based on Edge Finite-element Method

There is a desire to obtain rapid and stable inversion results and clearly reconstruct subsurface resistivity structure in frequency domain (FD) electromagnetics. Three-dimensional modeling of land FD controlled-source electromagnetic (CSEM) data is vital to improve the understanding of electromagnetic responses collected in increasingly complex geologic settings. Three-dimensional inversion of land FD-CSEM data is a mathematically non-unique problem with instability, due to the noise contained in the data and its inherent incompleteness. The main difference between our method and those from previous work is that the edge finite-element approach is applied to solve the three-dimensional FD-CSEM generated by a horizontal electric dipole source. Firstly, we formulate the edge finite-element equation through the Galerkin method, based on the Helmholtz equation of the electric fields. Secondly, in order to check the validity of the modeling code, we compare the numerical results with the analytical solutions for a homogeneous half-space model. For further tests, we calculate the electromagnetic responses for another two models with more practical structures. Finally, the three-dimensional inversion is carried out based on a regularization method with smoothness-constraints to obtain stable solutions.

Accelerating the T‐matrix approach to seismic full‐waveform inversion by domain decomposition

ABSTRACTA seismic variant of the distorted Born iterative inversion method, which is commonly used in electromagnetic and acoustic (medical) imaging, has been recently developed on the basis of the T‐matrix approach of multiple scattering theory. The distorted Born iterative method is consistent with the Gauss–Newton method, but its implementation is different, and there are potentially significant computational advantages of using the T‐matrix approach in this context. It has been shown that the computational cost associated with the updating of the background medium Green functions after each iteration can be reduced via the use of various linearisation or quasi‐linearisation techniques. However, these techniques for reducing the computational cost may not work well in the presence of strong contrasts. To deal with this, we have now developed a domain decomposition method, which allows one to decompose the seismic velocity model into an arbitrary number of heterogeneous domains that can be treated separately and in parallel. The new domain decomposition method is based on the concept of a scattering‐path matrix, which is well known in solid‐state physics. If the seismic model consists of different domains that are well separated (e.g., different reservoirs within a sedimentary basin), then the scattering‐path matrix formulation can be used to derive approximations that are sufficiently accurate but far more speedy and much less memory demanding because they ignore the interaction between different domains. However, we show here that one can also use the scattering‐path matrix formulation to calculate the overall T‐matrix for a large model exactly without any approximations at a computational cost that is significantly smaller than the cost associated with an exact formal matrix inversion solution. This is because we have derived exact analytical results for the special case of two interacting domains and combined them with Strassen's formulas for fast recursive matrix inversion. To illustrate the fact that we have accelerated the T‐matrix approach to full‐waveform inversion by domain decomposition, we perform a series of numerical experiments based on synthetic data associated with a complex salt model and a simpler two‐dimensional model that can be naturally decomposed into separate upper and lower domains. If the domain decomposition method is combined with an additional layer of multi‐scale regularisation (based on spatial smoothing of the sensitivity matrix and the data residual vector along the receiver line) beyond standard sequential frequency inversion, then one apparently can also obtain stable inversion results in the absence of ultra‐low frequencies and reduced computation times.

External inversion of thin-walled corrugated tubes

A thin-walled circular tube is known to be able to invert externally under axial compression when it is placed upon a profiled rigid die. Such a structure has become a promising candidate for energy absorbing applications due to its high reaction force and good load uniformity. However, the inversion process could become rather unstable when imperfections are present and the friction between the tube and die is high, which largely limits its application. In this paper, we propose the use of corrugated thin-walled tubes. Both simulation and experimental results show that thin-walled corrugated tubes can have a much more stable inversion in comparison with their circular counterparts. They are less sensitive to the surface imperfections, and no stringent requirement of lubrication on the die is necessary. Our research has paved the way for such devices to be used as an effective energy absorption device in many practical applications.

On the inverse problem of source reconstruction from coherence measurements.

We consider an inverse source problem for partially coherent light propagating in the Fresnel regime. The data are the coherence of the field measured away from the source. The reconstruction is based on a minimum residue formulation, which uses the authors' recent closed-form approximation formula for the coherence of the propagated field. The developed algorithms require a small data sample for convergence and yield stable inversion by exploiting information in the coherence as opposed to intensity-only measurements. Examples with both simulated and experimental data demonstrate the ability of the proposed approach to simultaneously recover complex sources in different planes transverse to the direction of propagation.

On the inverse problem of source reconstruction from coherence measurements

We consider an inverse source problem for partially coherent light propagating in the Fresnel regime. The data are the coherence of the field measured away from the source. The reconstruction is based on a minimum residue formulation, which uses the authors’ recent closed-form approximation formula for the coherence of the propagated field. The developed algorithms require a small data sample for convergence and yield stable inversion by exploiting information in the coherence as opposed to intensity-only measurements. Examples with both simulated and experimental data demonstrate the ability of the proposed approach to simultaneously recover complex sources in different planes transverse to the direction of propagation.

Monitoring of cyclic steam stimulation by inversion of surface tilt measurements

Temperature and pressure changes associated with the cyclic steam stimulation (CSS) used in heavy oil production from sands are accompanied by significant deformation. Inversion of induced reservoir pore-pressure changes from deformation measurements may provide a potentially powerful reservoir-monitoring tool if the issues of measurement noise, uncertainty in model parameterization, and numerical accuracy and stability can be resolved. We discuss inverting injection-induced reservoir pressure changes from observable surface deformations using a linear poroelastostatic model of a heavy oil reservoir. We also present results of inversion from surface tilt measurements taken at a production site undergoing CSS. We demonstrate that a stable inversion of the reservoir pore-pressure change can be achieved from sparse and noisy surface tilt measurements using constrained regularized optimization. The results provide an insight into the heterogeneity of reservoir stimulation and could help with optimizing well locations and stimulation protocols.

Retrieval of Elastodynamic Reflections From Passive Double‐Couple Recordings

AbstractVirtual Green's functions obtained by seismic interferometry (SI) can provide valuable reflectivity data that can complement tomographic inversion schemes. However, virtual reflections are affected by illumination irregularities that are typical of earthquake‐induced wavefields recorded by the receiver array. As a consequence, irregular source distributions, scattering (in case of suboptimal illumination), and complex source mechanisms can significantly disturb the retrieval of Green's function approximations by conventional SI methods. We introduce SI by full‐field multidimensional deconvolution (MDD) for elastodynamic wavefields as an alternative method to obtain body wave Green's functions under those typical circumstances. The advantage of this method compared to other MDD methods is that the kernel of its governing equation is exact. This alternative formulation of the kernel pertains to several advantages: the solution is less sensitive to artifacts and utilizes the free‐surface multiples in the data to estimate primary reflections. Moreover, the point spread function of the full‐field MDD method corrects more affectively for irregular illumination because it also addresses irregularities caused by scattering inside the medium. In order to compare full‐field MDD to existing SI methods, we model synthetic earthquake recordings in a subduction zone setting using an elastodynamic finite‐difference scheme with double couples of different orientations and peak frequencies. Our results show that SI by cross correlation suffers most under these circumstances. Higher‐quality reflections are obtained by the MDD methods, of which full‐field MDD involves the most stable inversion, and its results are least contaminated by artifacts.

Adaptive neural network output feedback control for flexible multi-link robotic manipulators

ABSTRACTIn this paper, a novel approach for adaptive control of flexible multi-link robots in the joint space is presented. The approach is valid for a class of highly uncertain systems with arbitrary but bounded dimension. The problem of trajectory tracking is solved through developing a stable inversion for robot dynamics using only joint angles measurement; then a linear dynamic compensator is utilised to stabilise the tracking error for the nominal system. Furthermore, a high gain observer is designed to provide an estimate for error dynamics. A linear in parameter neural network based adaptive signal is used to approximate and eliminate the effect of uncertainties due to link flexibilities and vibration modes on tracking performance, where the adaptation rule for the neural network weights is derived based on Lyapunov function. The stability and the ultimate boundedness of the error signals and closed-loop system is demonstrated through the Lyapunov stability theory. Computer simulations of the proposed robust controller are carried to validate on a two-link flexible planar manipulator.

Unexpected Occurrence of Mesospheric Frontal Gravity Wave Events Over South Pole (90°S)

AbstractSince 2010, Utah State University has operated an infrared Advanced Mesospheric Temperature Mapper at the Amundsen–Scott South Pole station to investigate the upper atmosphere dynamics and temperature deep within the vortex. A surprising number of “frontal” gravity wave events (86) were recorded in the mesospheric OH(3,1) band intensity and rotational temperature images (typical altitude of ~87 km) during four austral winters (2012–2015). These events are gravity waves (GWs) characterized by a sharp leading wave front followed by a quasi‐monochromatic wave train that grows with time. A particular subset of frontal gravity wave events has been identified in the past (Dewan & Picard, 1998) as “bores.” These are usually associated with wave ducting within stable mesospheric inversion layers, which allow them to propagate over very large distances. They have been observed on numerous occasions from low‐latitude and midlatitude sites, but to date, very few have been reported at high latitudes. This study provides new analyses of the characteristics of frontal events at high latitudes and shows that most of them are likely ducted. The occurrence of these frontal GW events over this isolated region strongly supports the existence of horizontally extensive mesospheric thermal inversion layers over Antarctica, leading to regions of enhanced stability necessary for GW trapping and ducting.

Quantifying the relationship between PM<sub>2.5</sub> concentration, visibility and planetary boundary layer height for long-lasting haze and fog–haze mixed events in Beijing

Abstract. Air quality and visibility are strongly influenced by aerosol loading, which is driven by meteorological conditions. The quantification of their relationships is critical to understanding the physical and chemical processes and forecasting of the polluted events. We investigated and quantified the relationship between PM2.5 (particulate matter with aerodynamic diameter is 2.5 µm and less) mass concentration, visibility and planetary boundary layer (PBL) height in this study based on the data obtained from four long-lasting haze events and seven fog–haze mixed events from January 2014 to March 2015 in Beijing. The statistical results show that there was a negative exponential function between the visibility and the PM2.5 mass concentration for both haze and fog–haze mixed events (with the same R2 of 0.80). However, the fog–haze events caused a more obvious decrease of visibility than that for haze events due to the formation of fog droplets that could induce higher light extinction. The PM2.5 concentration had an inversely linear correlation with PBL height for haze events and a negative exponential correlation for fog–haze mixed events, indicating that the PM2.5 concentration is more sensitive to PBL height in fog–haze mixed events. The visibility had positively linear correlation with the PBL height with an R2 of 0.35 in haze events and positive exponential correlation with an R2 of 0.56 in fog–haze mixed events. We also investigated the physical mechanism responsible for these relationships between visibility, PM2.5 concentration and PBL height through typical haze and fog–haze mixed event and found that a double inversion layer formed in both typical events and played critical roles in maintaining and enhancing the long-lasting polluted events. The variations of the double inversion layers were closely associated with the processes of long-wave radiation cooling in the nighttime and short-wave solar radiation reduction in the daytime. The upper-level stable inversion layer was formed by the persistent warm and humid southwestern airflow, while the low-level inversion layer was initially produced by the surface long-wave radiation cooling in the nighttime and maintained by the reduction of surface solar radiation in the daytime. The obvious descending process of the upper-level inversion layer induced by the radiation process could be responsible for the enhancement of the low-level inversion layer and the lowering PBL height, as well as high aerosol loading for these polluted events. The reduction of surface solar radiation in the daytime could be around 35 % for the haze event and 94 % for the fog–haze mixed event. Therefore, the formation and subsequent descending processes of the upper-level inversion layer should be an important factor in maintaining and strengthening the long-lasting severe polluted events, which has not been revealed in previous publications. The interactions and feedbacks between PM2.5 concentration and PBL height linked by radiation process caused a more significant and long-lasting deterioration of air quality and visibility in fog–haze mixed events. The interactions and feedbacks of all processes were particularly strong when the PM2.5 mass concentration was larger than 150–200 µg m−3.

A Modified Landweber Algorithm for Inversion of Particle Size Distribution Combined With Tikhonov Regularization Theory

Particle size distribution (PSD) measurement based on the static light scattering method has been widely used in the environmental field and combustion diagnostics, such as PM2.5 measurement and combustion process monitoring. The PSD inversion is mathematically related to the Fredholm integral equation of the first kind. Although the Tikhonov regularization algorithm is one of the effective inversion methods to solve the ill-posed linear equation, it still has the disadvantages of excessive smoothness and low accuracy. Thus, a preconditioned Landweber algorithm combined with the Tikhonov regularization theory (the improved algorithm) is proposed in this paper. The Tikhonov regularization theory is used to pretreat the preconditioner B contained in the preconditioned Landweber algorithm. Simulations are conducted to compare the inversion results of the conventional Landweber algorithm, the preconditioned Landweber algorithm, the Tikhonov Chahine algorithm, and the improved algorithm at different signal-to-noise ratios. A CCD-based small-angle forward scattering measurement system is built. A standardized polystyrene microsphere with a diameter of $35.05~\mu \text{m}$ is used to evaluate the above algorithms. Both numerical and experimental results show that the improved algorithm improves the accuracy of the inversion results and is insensitive to the ring parameter of the detector. The experimental results of the standardized polystyrene microsphere reveal that the relative errors for the median diameter $50~\mu \text{m}$ are better than 3%. The improved algorithm can provide a highly reliable and stable inversion result.

A direct method for nonlinear ill-posed problems

We propose a direct method for solving nonlinear ill-posed problems in Banach-spaces. The method is based on a stable inversion formula we explicitly compute by applying techniques for analytic functions. Furthermore, we investigate the convergence and stability of the method and prove that the derived noniterative algorithm is a regularization. The inversion formula provides a systematic sensitivity analysis. The approach is applicable to a wide range of nonlinear ill-posed problems. We test the algorithm on a nonlinear problem of travel-time inversion in seismic tomography. Numerical results illustrate the robustness and efficiency of the algorithm.