Rank-deficient Systems Research Articles

Paving the Way for Consensus: Convergence of Block Gossip Algorithms

Gossip protocols are popular methods for average consensus problems in distributed computing. We prove new convergence guarantees for a variety of such protocols, including path, clique, and synchronous pairwise gossip. These arise by exploiting the connection between these protocols and the block randomized Kaczmarz method for solving linear systems. Moreover, we extend existing convergence results for block randomized Kaczmarz to allow for a more general choice of blocks, rank-deficient systems, and provide a tighter convergence rate guarantee. Previous results for block randomized Kaczmarz assumed blocks formed a paving of the systems; our analysis, contrary to the title of our paper, generalizes to a broader class of blocks that we call coverings. We furthermore apply this analysis to inconsistent consensus models and obtain similar guarantees. An extensive empirical analysis of these methods is provided for a variety of synthetic and real networks.

Sensor fault detection and isolation via networked estimation: rank-deficient dynamical systems

This paper considers model-based fault detection of large-scale (possibly rank-deficient) dynamic systems. Assuming only global (and not local) observability over a sensor network, we introduce a single time-scale networked estimator/observer. Sensors take local outputs/measurements of system states with partial observability and share their information (including estimation and/or output) over a communication network, and gain distributed observability. We define the conditions on the network structure ensuring distributed observability and stabilising the error dynamics. However, system outputs are prone to faults and uncertainties, which affect the state estimation of all sensors as a consequence of communicating (possibly) faulty data. From the cyber-physical-systems (CPS) perspective, such faults add bias to the data transferred from the physical layer (dynamic system) to the cyber layer (sensor network). In this work, we propose a localised fault detection and isolation (FDI) mechanism at sensors to secure distributed estimation. This protocol enables every sensor to locally identify the possible fault at the sensor measurement, and, via local detection and isolation, to prevent the spread of biased/faulty information over the network. This distributed isolation and localisation of fault follows from our partial observability assumption instead of full observability at every sensor. Then, other sensors can estimate/track the system by using observationally-equivalent output information to recover for possible loss of observability. In particular, we study rank-deficient systems as they are known to demand more information-sharing, and thus, are more vulnerable to the spread of possible faults over the network. One challenge is the detection of faults in the presence of system/output noise without making (simplifying and unrealistic) upper-bound assumptions on the noise support. We resolve this by adopting probabilistic threshold designs on the residuals. Further, we show that additive faults at rank-deficiency-related outputs affect the residuals at all sensors, a consequence that mandates more constraints on the (distributed) FDI strategy. We address this problem by constrained LMI design of the feedback gain matrix. Finally, we design q-redundant distributed estimators, resilient to isolation/removal of up to q number of faulty sensors, and further, we consider thresholding residual history over a sliding time-window, known as the stateful FDI.

Approaches for handling high-dimensional cluster expansions of ionic systems

Disordered multicomponent systems attract great interest due to their engineering design flexibility and subsequent rich space of properties. However, detailed characterization of the structure and atomic correlations remains challenging and hinders full navigation of these complex spaces. A lattice cluster expansion is one tool to obtain configurational and energetic resolution. While in theory a cluster expansion can be applied to any system of any dimensionality, the method has primarily been used in binary systems or ternary alloys. Here we apply cluster expansions in high-component ionic systems, setting up the largest cluster expansion ever attempted to our knowledge. In doing so, we address and discuss challenges specific to high-component ionic systems, namely charge state assignments, structural relaxations, and rank-deficient systems. We introduce practical procedures to make the fitting and analysis of complex systems tractable, providing guidance for future computational studies of disordered ionic systems.

Internal Model Control for Rank-Deficient System with Time Delays Based on Damped Pseudo-Inverse

It is a challenge to design a satisfactory controller for a complex multivariable industrial system with minimal offsetting and a slow response. An internal model control method is proposed for rank-deficient systems with a time delay based on a damped pseudo-inverse. An internal model control was designed to obtain the desired dynamic characteristics of the system by transforming the time-delay system into a system without a time delay, following the Pade approximation approach. By introducing a damping factor, the internal model controller was designed based on a damped pseudo-inverse, since the inverse matrix of the rank-deficient system does not exist. Furthermore, a singular value decomposition was used to analyze the steady-state performance of the system. The selection of the damping factor was also presented, and a μ analysis was made to evaluate the stability of the system. To demonstrate the effectiveness of the proposed method, a crude distillation process with five inputs and four outputs was considered as an example. The simulation results illustrate that not only can the proposed strategy guarantee the system’s stability, but it also has a relatively good dynamic performance.

Internal model control for structured rank deficient system based on full rank decomposition

In this paper, an internal model control method is first proposed for structured rank deficient systems based on full rank decomposition. The system is first converted into a column full rank system by designing a pre-compensator. Then a feedback-compensator is designed to improve the dynamic characteristics of the full rank system and decrease the controller design difficulties. Rather than performing complex designing calculations, the pre- and feedback- compensators are designed by the full rank decomposition method. Furthermore, the non-square relative gain subsystem selection criterion is used to choose the square subsystem and to realize loop pairing. Consequently, the selected square subsystem is used as an internal model to design the internal model controller. Finally, a simple process is taken as the simulation object to demonstrate the validity and feasibility of the new method. Simulations results illustrate that the proposed strategy is not only simple and easy to implement but also has a good performance even the system model is mismatched.

A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.

Joint Quantum-Assisted Channel Estimation and Data Detection

Joint channel estimation (CE) and multi-user detection (MUD) have become a crucial part of iterative receivers. In this paper, we propose a quantum-assisted repeated weighted boosting search (QRWBS) algorithm for CE and we employ it in the uplink of multiple-input multiple-output orthogonal frequency division multiplexing systems, in conjunction with the maximum a posteriori probability (MAP) MUD and a near-optimal quantum-assisted MUD (QMUD). The performance of the QRWBS-aided CE is evaluated in rank-deficient systems, where the number of receive antenna elements (AEs) at the base station (BS) is lower than the number of supported users. The effect of the channel impulse response prediction filters, of the power delay profile of the channels, and of the Doppler frequency on the attainable system performance is also quantified. The proposed QRWBS-aided CE is shown to outperform the RWBS-aided CE, despite requiring a lower complexity, in systems where iterations are invoked between the MUD, the CE, and the channel decoders at the receiver. In a system, where $U=7$ users are supported with the aid of $P=4$ receive AEs, the joint QRWBS-aided CE and QMUD achieves a 2-dB gain, when compared with the joint RWBS-aided CE and MAP MUD, despite imposing 43% lower complexity.

A Performance Study of Semidefinite Relaxation Detector in Spatially Correlated and Rank Deficient Large MIMO Systems

Large MIMO detection has gained significant attention in the recent past with computational complexity as the research focus. However, they assume the channel to be i.i.d and uncorrelated, which is not a valid assumption in practice due to the fixed physical space constraints in large MIMO. Nevertheless, there is a little work carried out in these lines. In this paper, we consider the problem of detection in large spatial multiplexing MIMO systems and we investigate the semidefinite relaxation (SDR) approach to solve this problem. We investigate the applicability of SDR approach in large MIMO setting and study its performance in spatially correlated and rank deficient channel conditions. Through the simulation results, we demonstrate the superior performance of semidefinite relaxation detector over other existing methods in uncorrelated and correlated large MIMO systems especially in low SNR regime. The performance of SDR detector is noteworthy with large number of antennas despite the system being rank deficient and the average running time also scales up well for large systems.

Bicoprime Factor Stability Criteria and Uncertainty Characterisation

Bicoprime factorisations can be considered a generalisation of the well known coprime factorisations. This paper deals with the possible applications of such factorisations in control theory. The concept of minimal dimension bicoprime factorisations is introduced and shown to be especially beneficial in the case of normal rank deficient and redundant control systems. Some methods of obtaining a bicoprime factorisation for a plant are given based on state-space data, right or left coprime factorisations and left or right standard factorisations. Finally, bicoprime factor uncertainty is characterised and shown to have an interesting structure closely resembling the standard four-block uncertainty.

Stopping Distance from Observations by Stepwise Regression

AbstractIn a regression equation with many terms stepwise regression allows to pick out significant terms and to exclude insignificant ones. This can be used to solve rank deficient equation systems. Yet significance is not condition. There is a rather simple method for stepwise regression starting from extended normal equations proposed by Efroymson 1960 [1] and improved by Breaux 1968 [2]. Numerically more stable methods apply orthogonal transformations (Eldén 1972 [3], Gragg‐LeVeque‐Trangenstein 1979 [4]). If symmetry is properly made use of [2, 5] the method of Efroymson is the most efficient one with respect to computing time. But reasonable results are obtained only if the terms in the regression equation are physically justified. This illustrates the search for a formula of the stopping distance derived from observations, an exercise given in [9], a book widely used on some universities. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On an iterative method for solving the least squares problem of rank-deficient systems

In this paper, we first show an iterative method for finding the least squares (LS) solution to the inconsistent system Ax=b, where A is an m×n matrix of rank r. The method is an iteration scheme for consistent system of linear equations M=bˆ which is an augmented system associated with Ax=b. It denotes that under some conditions, the sequence, 0, 1, 2, … , converges to the LS solution of the system Ax=b for every initial vector 0, where (M+γ E)i=γ Ei−1+bˆ, for i=1, 2, … . In our numerical test, we propose to find E without using decomposition methods. The improved timings are shown with matrices of substantial size.

Solving Rank-deficient Linear Systems for the Estimation of the Atmospheric Phase Delay Parameter

For each pixel, the values of the unwrapped phase produced by interferometric pairs can be parameterized and phase components, such as height, atmospheric path delay difference for the dates of every interferometric set image acquisition, and unwrapping error can be estimated through the solution of a rank deficient system of equations. There are several state of the art methods for solving rank deficient systems of equations, such as the Lower-Upper method, QR factorization and Singular Value Decomposition (SVD). In a previous work, the improved SVD method, which enables the accurate estimation of phase components, has been proposed. In this study, one alternative method for the estimation of the differential interferometry parameters, the improved QR, is proposed. The method relies on the reorganization of the system of equations by using external meteorological data which correspond to the acquisition dates that are less present in the interferometric SAR data set. All the aforementioned methods were tested and compared. The improved SVD and improved QR methods yield the most satisfactory results for all the phase components. The former estimates the heights by achieving almost GPS accuracy, whereas the latter is more robust, producing results of almost the same accuracy for all the estimated atmospheric path delays. However, the requirement of known values for at least two phase components makes the improved QR method less operational.

Convex optimization under inequality constraints in rank-deficient systems

Many geodetic applications require the minimization of a convex objective function subject to some linear equality and/or inequality constraints. If a system is singular (e.g., a geodetic network without a defined datum) this results in a manifold of solutions. Most state-of-the-art algorithms for inequality constrained optimization (e.g., the Active-Set-Method or primal-dual Interior-Point-Methods) are either not able to deal with a rank-deficient objective function or yield only one of an infinite number of particular solutions. In this contribution, we develop a framework for the rigorous computation of a general solution of a rank-deficient problem with inequality constraints. We aim for the computation of a unique particular solution which fulfills predefined optimality criteria as well as for an adequate representation of the homogeneous solution including the constraints. Our theoretical findings are applied in a case study to determine optimal repetition numbers for a geodetic network to demonstrate the potential of the proposed framework.

Robust Fixed-Complexity Sphere Decoders for Rank-Deficient MIMO Systems

Sphere decoder (SD) has variable complexity, and the traditional fixed-complexity sphere decoder (FSD) is not applicable for rank-deficient (NT > NR) multiple input multiple output (MIMO) systems (i.e. the down link detection in MIMO systems, or systems with highly correlated MIMO channels). To overcome these difficulties, in this paper, robust fixed-complexity sphere decoders (RFSD-s) based on new preprocessing algorithms are proposed for rank-deficient MIMO systems. With respect to the cases without and with information on the level of noise, RFSD using zero-forcing technique (RFSD-ZF) and FSD using minimum mean square error technique (FSD-MMSE) are proposed respectively. To reduce the computational complexity of RFSD-ZF, a simplified RFSD-ZF (SRFSD-ZF) with little performance loss is also introduced, the theoretical proof is given to support the feasibility of SRFSD-ZF. Simulation results show that, besides better performance than the traditional FSD, the proposed techniques are robust to the configuration of MIMO antennas (both NT ≤ NR and NT > NR), which is another advantage over the traditional FSD.

Generalized MBER-Based Vector Precoding Design for Multiuser Transmission

We propose a generalized vector precoding (VP) design based on the minimum bit error rate (MBER) criterion for multiuser transmission in the downlink of a multiuser system, where the base station (BS) equipped with multiple transmitting antennas communicates with single-receiving-antenna mobile station (MS) receivers each having a modulo device. Given the knowledge of the channel state information and the current information symbol vector to be transmitted, our scheme directly generates the effective symbol vector based on the MBER criterion using the particle swarm optimization (PSO) algorithm. The proposed PSO-aided generalized MBER VP scheme is shown to outperform the powerful minimum mean-square-error (MMSE) VP and improved MMSE-VP benchmarks, particularly for rank-deficient systems, where the number of BS transmitting antennas is lower than the number of MSs supported.

A Transceiver Design Based on Uniform Channel Decomposition and MBER Vector Perturbation

Uniform channel decomposition (UCD), as an improvement of geometric mean decomposition (GMD), is capable of decomposing a multiple-input-multiple-output (MIMO) channel into multiple subchannels having an identical capacity. Vector precoding (VP) is a powerful scheme, which is capable of mitigating multiuser interference (MUI) at the transmitter, provided that the channels of the users are known. In this paper, a novel joint transceiver design based on the UCD and the minimum bit error rate (MBER) VP principles is proposed for MIMO systems, in which the precoding and equalization matrices are calculated by UCD, whereas the perturbation vector is chosen to minimize the system's bit error ratio (BER). This UCD-MBER-VP transceiver design outperforms several state-of-the-art benchmark algorithms in terms of achievable BER performance without imposing an increased computational complexity, particularly for rank-deficient systems.

Downlink MBER Transmit Beamforming Design Based on Uplink MBER Receive Beamforming for TDD-SDMA Induced MIMO Systems

The downlink minimum bit error rate (MBER) transmit beamforming is directly designed based on the uplink MBER receive beamforming solution for time division duplex (TDD) space-division multiple-access (SDMA) induced multiple-input multiple-output (MIMO) systems, where the base station (BS) is equipped with multiple antennas to support multiple single-antenna mobile terminals (MTs). It is shown that the dual relationship between multiuser detection and multiuser transmission can be extended to the rank-deficient system where the number of users supported is more than the number of transmit antennas available at the BS, if the MBER design is adopted. The proposed MBER transmit beamforming scheme is capable of achieving better performance over the standard minimum mean square error transmit beamforming solution with the support of low-complexity and high power-efficient MTs, particularly for rank-deficient TDD-SDMA MIMO systems. The robustness of the proposed MBER transmit beamforming design to the downlink and uplink noise or channel mismatch is investigated using simulation.

Three-Stage Irregular Convolutional Coded Iterative Center-Shifting $K$-Best Sphere Detection for Soft-Decision SDMA–OFDM

The iterative exchange of extrinsic information between the K-best sphere detector (SD) and the channel decoder is appealing since it is capable of achieving a near maximum a posteriori (MAP) performance at a moderate complexity. However, the computational complexity imposed by the K-best SD significantly increases when using a large value of K for the sake of maintaining a near-MAP performance in a high-throughput uplink spatial-division multiple access (SDMA) orthogonal frequency-division multiplexing (OFDM) system supporting a large number of users and/or a high number of bits/symbol. This problem is further aggravated when the number of users/mobile stations (MSs) U exceeds that of the receive antennas N at the base station (BS), namely, in the challenging scenario of rank-deficient systems. We demonstrate that the iterative decoding convergence of this two-stage system may be improved by incorporating a unity rate code (URC) having an infinite impulse response, which improves the efficiency of the extrinsic information exchange. Although this results in a slightly more complex three-stage system architecture, it allows us to use a low-complexity SD having a significantly reduced candidate list size N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cand</sub> . Alternatively, a reduced signal-to-noise ratio (SNR) is required. For example, given a target bit error ratio (BER) of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> and N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cand</sub> = 32 for the SD, the three-stage receiver is capable of achieving a performance gain of 2.5 dB over its two-stage counterpart in a rank-deficient SDMA/OFDM 4-quadratic-amplitude modulation (4-QAM) system supporting U = 8 cochannel users and employing N = 4 receive antennas at the BS, namely, in an (8 x 4) system. For the sake of further enhancing the three-stage concatenated receiver, the proposed iterative center-shifting SD scheme and the irregular convolutional codes (IrCCs) are intrinsically amalgamated, which leads to an additional performance gain of 2 dB.

Fast solving of weighted pairing least-squares systems

This paper presents a generalization of the “weighted least-squares” (WLS), named “weighted pairing least-squares” (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual “Cholesky factorization-backward substitution” method for solving linear systems.

Multiuser MIMO OFDM Based TDD/TDMA for Next Generation Wireless Communication Systems

Multiple-input multiple-output (MIMO) wireless technology in combination with orthogonal frequency division multiplexing (MIMO OFDM) is an attractive air-interface solution for next-generation wireless local area networks (WLANs), wireless metropolitan area networks (WMANs), and fourth-generation mobile cellular wireless systems. In this paper, one multiuser MIMO OFDM systems with TDD/TDMA was proposed for next-generation wireless mobile communications, i.e., TDD/TDMA 4G, which can avoid or alleviate the specific limitations of existing techniques designed for multiuser MIMO OFDM systems in broadband wireless mobile channel scenarios, i.e., bad performance and extreme complexity of multiuser detectors for rank-deficient multiuser MIMO OFDM systems with CDMA as access modes, extreme challenges of spatial MIMO channel estimators in rank-deficient MIMO OFDM systems, and exponential growth complexity of optimal sub-carrier allocations for OFDMA-based MIMO OFDM systems. Furthermore, inspired from the Steiner channel estimation method in multi-user CDMA uplink wireless channels, we proposed a new design scheme of training sequence in time domain to conduct channel estimation. Training sequences of different transmit antennas can be simply obtained by truncating the circular extension of one basic training sequence, and the pilot matrix assembled by these training sequences is one circular matrix with good reversibility. A novel eigenmode transmission was also given in this paper, and data symbols encoded by space---time codes can be steered to these eigenmodes similar to MIMO wireless communication systems with single-carrier transmission. At the same time,, an improved water-filling scheme was also described for determining the optimal transmit powers for orthogonal eigenmodes. The classical water-filling strategy is firstly adopted to determine the optimal power allocation and correspondent bit numbers for every eigenmode, followed by a residual power reallocation to further determine the additional bit numbers carried by every eigenmode. Compared with classical water-filling schemes, it can also obtain larger throughputs via residual power allocation. At last, three typical implementation schemes of multiuser MIMO OFDM with TDMA, CDMA and OFDMA, i.e., TDD/TDMA 4G, VSF-OFCDM and FuTURE B3G TDD, were tested by numerical simulations. Results indicated that the proposed multiuser MIMO OFDM system schemes with TDD/TDMA, i.e., TDD/TDMA 4G, can achieve comparable system performance and throughputs with low complexity and radio resource overhead to that of DoCoMo MIMO VSF-OFCDM and FuTURE B3G TDD.