Linear Systems Research Articles

A State Estimation Method for Rectangular Singular Systems Based on State Decomposition

This paper proposes a state estimation method of linear discrete rectangular singular systems. The system is observable and regular, and the system matrix is rectangular without full column rank. To give the estimation of the state, the state is decomposed into two parts based on QR factorization, and the weighted least squares method is used to obtain the prediction of one part of the state. Then, the partial measurement equation is used to obtain the prediction of the other part of the state, and the projection theorem is used to obtain the state estimation value. Combined with the data-driven idea, a Kalman filtering algorithm based on historical data modeling is established. Finally, the feasibility and effectiveness of our approach is discussed and verified through performance analysis and numerical simulation perspectives.

Susceptibility of two different PCA pumps to inaccurate delivery associated with pole position at low flow-rates in a pediatric setting - an experimental study.

Vertical displacement of infusion pumps used in patient-controlled analgesia can cause irregularities in drug delivery and is especially crucial at low flow rates, which are commonly used in pediatrics. There is only scarce data available regarding the extent of these inaccuracies. The current in vitro study therefore aimed at a comparison of the performance of two commonly used PCA pumps at different pole positions due to vertical displacement. The Syramed® µSP6000 Chroma syringe infusion pump featuring a stepper motor drive was compared to the CADD®-Solis pump utilizing a linear peristaltic pump system at two different flow rates (0.3ml/h and 1ml/h) and three different levels of height (0, + 50 and - 50cm). Flow patterns and delivered volumes were measured after every change in position and infusion boluses, retrograde aspiration volumes and zero-drug delivery time were calculated. The Syramed® pump was more susceptible to vertical displacement than the CADD®-Solis pump and showed overall greater inaccuracies in the delivered volumes as well as higher infusion boluses, retrograde aspiration volumes and zero-drug delivery time at both flow rates. The observed differences between the pumps might be explained by the higher compliance of this syringe pump and the diverse working mechanisms. Overall, the CADD®-Solis pump might be considered a preferable option for patient-controlled analgesia in children. It is nonetheless essential for medical staff to be aware of the effects of vertical displacement of PCA pumps and to minimize these displacements as much as possible.

Controllability and minimum energy control problem of two-dimensional continuous-time fractional linear systems

The effectiveness of this paper lies in a new formulation and solution of the minimum energy control problem for the fractional two-dimensional systems described by the Fornasini–Marchesini first models. The problem of controllability and minimum energy control of a general fractional two-dimensional Fornasini–Marchesini model is newly considered by the authors in this work. Necessary and sufficient controllability conditions are then established using the Gramian matrix. To carry out this study, a new test was described and used in order to give the existence conditions of the solution to the minimum energy control problem and the procedures for calculating an input minimising the given performance index. The usefulness and the precision of the proposed procedure is demonstrated by a numerical examples, Darboux equation and metal rolling process model as a practical application.

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.

A note on the approximate controllability of damped second-order differential inclusions with state-dependent delay

In this paper, we use Bohnenblust–Karlin's fixed point theorem and the theory of strongly continuous cosine families of bounded linear operators to study the approximate controllability of damped second-order differential inclusions with state-dependent delay in Banach spaces. In particular, we obtain a new set of sufficient conditions for the approximate controllability of damped second-order differential inclusions with state-dependent delay under the assumption that the corresponding linear system is approximately controllable. An example is provided to illustrate our main results.

A fast block sparse Kaczmarz algorithm for sparse signal recovery

The randomized sparse Kaczmarz (RSK) method is an iterative algorithm for computing sparse solutions of linear systems. Recently, Tondji and Lorenz analyzed the parallel version of the RSK method and established its linear expected convergence by implementing a randomized control scheme for subset selection at each iteration. Expanding upon this groundwork, we explore a natural extension of the randomized control scheme: greedy strategies such as the Motzkin criteria. Specifically, we propose a fast block sparse Kaczmarz algorithm based on the Motzkin criterion. It is proved that the proposed method converges linearly to the sparse solutions of the linear systems. Additionally, we offer error estimates for linear systems with noisy right-hand sides, and show that the proposed method converges within an error threshold of the noise level. Numerical results substantiate the feasibility of our proposed method and highlight its superior convergence rate compared to the parallel version of the RSK method.

On the Method for Constructing Null-Controllable Sets for Linear Discrete-Time Systems with Summary Control Constraints

On the Method for Constructing Null-Controllable Sets for Linear Discrete-Time Systems with Summary Control Constraints

Quantum algorithm for dynamic mode decomposition integrated with a quantum differential equation solver

We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O(polylogN) for an N-dimensional system. This is an exponential speedup over known classical algorithms with at least O(N) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Output feedback fault-tolerant Q-learning for discrete-time linear systems with actuator faults

This paper presents an innovative output feedback fault-tolerant Q-learning algorithm that can be implemented online without relying on explicit system models or fault details. In the face of actuator faults, finding optimal Fault-Tolerant Control (FTC) solutions that can stabilize the faulty system poses significant challenges. The proposed approach is implemented online without the need for system dynamics and actuator fault information. Furthermore, it operates without relying on full state measurements, utilizing only the input-output data of the faulty system. An innovative representation of the output feedback Fault-Tolerant Q-function (FTQF) is established using input-output data. Subsequently, a model-free optimal output feedback FTC policy is obtained from the developed FTQF. Then, a fault-tolerant Q-learning algorithm is formulated to iteratively acquire the optimal FTC policy in real-time, eliminating the necessity for system dynamics and actuator fault information. The proposed algorithm exhibits immunity to excitation noise bias, even without considering a discounting factor. Furthermore, the proposed Q-learning approach is proven to be effective in stabilizing faulty closed-loop system. Finally, the proposed algorithm's efficiency is validated through numerical simulations of F-16 autopilot aircraft dynamics.

Non-Fragile Sampled Control Design for an Interconnected Large-Scale System via Wirtinger Inequality

A control design for a linear large-scale interconnected system composed of identical subsystems is presented in this paper. The control signal of all subsystems is sampled. For different subsystems, the sampling times are not identical. Nonetheless, it is assumed that a bound exists for the maximal sampling time. The control algorithm is designed using the Wirtinger inequality, and the non-fragile control law is proposed. The size of the linear matrix inequalities to be solved by the proposed control algorithm is independent of the number of subsystems composing the overall system. Hence, the algorithm is computationally effective. The results are illustrated by two examples. The first example graphically illustrates the function of the proposed algorithm while the second one compares with a method for stabilizing a large-scale system obtained earlier, thus illustrating the improved capabilities of the presented algorithm.

Optical and mechanical squeezing with coherent feedback control beyond the resolved-sideband regime

The noise reduction in squeezed states makes them a powerful nonclassical resource in quantum technologies, especially in quantum sensing, and an essential step for the ongoing development of non-Gaussian states. In particular, optical and mechanical squeezing can both be generated when light interacts with a mechanical oscillator. Here we use a linear optical system as a controller and use coherent optical feedback to enhance squeezing in the unresolved-sideband regime of optomechanics. Feedback control is generally profitable for cooling, manipulating, and stabilizing the dynamics of mechanical oscillators; coherent feedback is then notable for its ability to steer a quantum system toward the highest squeezing without measurement, avoiding backaction. We thoroughly maximize optical and conditional mechanical squeezing feasible with state-of-the-art experimental setups and derive the optimal conditions under which quantum squeezing is generated in the system. Published by the American Physical Society 2024

An adaptive internal mass source wave-maker for short wave generation

The mass source wave-maker is commonly employed for generating water waves in numerical simulations, during which a correct amount of mass is introduced or subtracted from the internal flow region to produce target waves. The method has proven to be effective in producing waves in shallow and intermediate water depths, while its efficiency is declined for short wave generation. The main reason for this efficiency declination is that the internal mass source in deeper water region is not effective to generate short waves with their motions primarily on water surface. In order to overcome this shortcoming, many of the previous numerical treatments have introduced various enhancement factors into the source functions, which are empirically obtained and also violate the law of mass conservation. In this study, we develop a new adaptive internal wave-maker model that can be self-adjusted to suit different wave conditions. The line source starts from the bottom and extends to the computational cell right beneath free surface at each time step. The depth dependent weighting coefficient is introduced to the source function based on the linear wave theory for each wave component. No empirical coefficients are necessary, and the mass conservation is strictly and explicitly enforced. In principle, the method can be applied to all types of linear waves in the entire range of kh. The numerical experiments show that the present method can produce very good results for linear waves with kh up to 16.11, adequate for most of wave conditions in coastal engineering. For generation of fifth-order Stokes waves, the method can be extended straightforwardly for each of five wave components. For irregular waves composed of many linear wave components, different weighting coefficients can be readily calculated for each of them, respectively. As a result, the new model can generate irregular waves with overall better performance of reproducing wave spectrum, whose high-frequency part has been underestimated by previous methods. The numerical experiments also show that the new model can produce better results for focused waves where many linear waves of different frequencies start from the same point with specific phase angles, due to its capability of generating shorter wave components.

Quantum positive matrix-positive matrix multiplication algorithm

Abstract In quantum computing,
matrix multiplication occurs as a subroutine in more complex algorithms. 
We improve the quantum linear system algorithm 
[Harrow et al. 2009 Phys. Rev. Lett. 103 150502] 
to perform sparse matrix-matrix multiplication and propose a subroutine for parallel vector addition. 
Based on them, a quantum algorithm is exhibited for N×N positive matrix-positive matrix multiplication.
For some small ratio of the maximum element to the minimum element of the positive matrices (i.e.,O(poly logN)), 
this algorithm achieves exponential acceleration.

Stability Characteristics of Natural Gas Hydrate Wellbores Based on Thermo-Hydro-Mech Modeling

During drilling operations, drilling fluids undergo heat exchange with hydrate-bearing formations. The intrusion of drilling fluids affects hydrate stability, leading to variations in stress fields around a wellbore and complex scenarios such as borehole collapse, significantly hindering the efficient development of natural gas hydrate resources. This study establishes a thermo-hydro-mech model for hydrate-bearing inclined wells based on linear thermoelastic porous media theory and an appropriately high inhibitive drilling fluid temperature. This research reveals that drilling should follow directions of minimum horizontal stress or perpendicular to maximum horizontal stress during drilling operations to control wellbore stability. Hydrate decomposition due to factors like drilling fluid pressure and temperature can rapidly reduce the strength of low-saturation formations, significantly increasing the risk of wellbore instability. Therefore, selecting appropriate highly inhibitive and low-temperature drilling fluids during drilling operations helps control hydrate decomposition and reduce fluid intrusion, thereby mitigating risks associated with wellbore instability.

Kernel multigrid on manifolds

Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a τ≥2-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.

Nonlinear normal modes and response to random inputs of systems with bilinear stiffness

This work seeks to provide a comprehensive review of the effects that stiffness bilinearity can have on the nonlinear modes of a system, its response in a random vibration environment, and the connection between the two. Stiffness bilinearity here refers to a continuous piecewise linear force vs displacement function that is composed of two linear regions. This work focuses on a bilinear stiffness function that is regularized so there is a smooth transition at the point where the two linear regions meet. A single-degree-of-freedom system (SDOF) and a two-degree-of-freedom (2DOF) system are explored and several interesting behaviors are shown. The SDOF bilinear spring model is characterized by four parameters: the low amplitude frequency, the ratio of the linear stiffnesses on either side of the transition, the displacement at which the transition occurs, and the rate or sharpness of the transition. The effect of each parameter on the shape of the NNM is described. In a 2DOF system, these parameters have similar effects, but modal coupling is found to play a significant role. When a bilinear system is subjected to random excitation, many harmonics appear in the response for both the SDOF and 2DOF cases. The root-mean-square (RMS) response of the bilinear system can be larger or smaller than the corresponding linear case depending on the values of the parameters and the type of forcing (broadband, bandlimited, in the shape of a vibration mode, etc.). However, many cases were observed in which the RMS response of the bilinear system was almost the same as that of a linear system, and hence the response could be predicted well using linear analysis. It is hoped that the results presented herein can assist engineers in helping to determine when linear analysis would be adequate to predict the failure of a system, when a rigorous nonlinear analysis is required, and what phenomena are likely to be observed in the latter case.

Curves on Brill–Noether special K3 surfaces

AbstractMukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface is equivalent to the Brill–Noether generality of smooth curves in the linear system , is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus , which provides a new and unified proof even in the genera where Mukai models exist.

Relationship Between High Frequency Component of Heart Rate Variability and Delta EEG Power During Sleep in Women With Irritable Bowel Syndrome Compared to Healthy Women.

To explore the relationship between the high frequency (HF) heart rate variability (HRV) and electroencephalogram (EEG) delta band power in women with irritable bowel syndrome (IBS) versus healthy control women. Twenty women with IBS and twenty healthy controls were studied over three consecutive nights using polysomnography in a sleep laboratory. To avoid the first night effect, only second-night data were analyzed. Power spectral analysis was applied to HRV and EEG recordings. The linear system coherence/phase analysis assessed the relationship between normalized HF power of HRV and normalized delta band power of EEG during the first four NREM-REM sleep cycles. Women with IBS exhibited a significantly higher percentage of NREM sleep, higher normalized HF, lower normalized low frequency (LF) and decreased LF/HF ratio of HRV in the first four NREM-REM sleep cycles compared to controls. Additionally, their normalized delta band power was significantly lower in these sleep cycles and over the whole night. The phase shift between HF and delta band power was significantly longer in the IBS group. While the coherence between normalized HF and normalized delta band power was lower in the IBS group, the difference was not statistically significant. The coherence/phase analysis showed a dysregulated interaction between autonomic and central nervous systems in women with IBS, manifested by increased lag time between cardiac and EEG delta band power compared to healthy controls. Whether this dysregulation contributes to the pathophysiology of IBS remains to be determined.

Three-dimensional airborne electromagnetic forward modeling based on multiscale hexahedral finite-element method

Slow forward modeling is the main factor that restricts the practical use of three-dimensional (3D) inversion and interpretation of airborne electromagnetic (AEM) data. To improve the modeling efficiency in 3D AEM, we propose a new multiscale finite-element (MsFE) method based on unstructured hexahedral meshes. Compared with the traditional 3D AEM forward modeling, the main advantage of our newly developed method is that it can simulate complex underground structures in the earth quickly. Since we can fit the earth's topography or the anomalous bodies underground using a small number of hexahedral grids, we can quickly model them using MsFE. The main idea of the MsFE forward modeling method is to construct an interpolation operator between a coarse and a dense mesh and use the interpolation operator to map the conventional FE coefficient matrix to the MsFE coefficient matrix and thus reduce the number of unknowns in the modeling process. This will vastly reduce the scale of the linear equations system. We validate our method by simulating a typical mountain peak model and demonstrate its effectiveness by simulating numerous synthetic models and a model from Voisey Bay's Ovoid sulfide deposit, Canada.