Minimum Condition Number Research Articles

Robust Exact Pole Placement via an LMI-Based Algorithm

This technical note deals with the robust exact pole placement problem: pole placement algorithms that guarantee a small variation of the assigned poles against possible perturbations. The solution to this problem is related to the solvability of a Sylvester-like equation. Thus, the main issue is to compute a well-conditioned solution to this equation. Also, the best candidate solution must possess the minimal condition number, to reduce sensitivity to perturbation. This problem is cast as a global optimization under linear matrix inequality constraints, for which a numerical convergent algorithm is provided and compared with the most attractive methods in the literature.

Integrated Design for Configuration/Vibration Control of Hexapod Platform

The Hexapod configuration and vibration control integrated design is treated as an optimal problem of choosing minimum control energy index among a family of Hexapod with optimal Jacobian condition number. Based on the theory of optimal Jacobian condition number, a programming problem is given, which takes control parameter of nonlinear feedback adaptive disturbance canceller(ADC) method, radius of Hexapod base plate and half separation angle between adjacent actuator attachment points on the up plate as design variables, control convergence condition, workspace, and space between attachment points as constrained conditions, and weighted sum of six actuators’ average energy consumption and its variance component as objective function. As the optimal model and its sensitivities contain implicit functions of design variables, the primal problem is approached by 2-rank elliptical response surface method with D-optimal test design. The original programming problem is settled by solving sequence of approximate problems. The example result gives out rules and key points of controller and configuration design of Hexapod, and demonstrates validity and feasibility of the proposed method. Compared with original developed Hexapod, the optimized Hexapod has the minimum condition number, and the advantages in control energy consumption and distribution.

FRAPP: a framework for high-accuracy privacy-preserving mining

To preserve client privacy in the data mining process, a variety of techniques based on random perturbation of individual data records have been proposed recently. In this paper, we present FRAPP, a generalized matrix-theoretic framework of random perturbation, which facilitates a systematic approach to the design of perturbation mechanisms for privacy-preserving mining. Specifically, FRAPP is used to demonstrate that (a) the prior techniques differ only in their choices for the perturbation matrix elements, and (b) a symmetric positive-definite perturbation matrix with minimal condition number can be identified, substantially enhancing the accuracy even under strict privacy requirements. We also propose a novel perturbation mechanism wherein the matrix elements are themselves characterized as random variables, and demonstrate that this feature provides significant improvements in privacy at only a marginal reduction in accuracy. The quantitative utility of FRAPP, which is a general-purpose random-perturbation-based privacy-preserving mining technique, is evaluated specifically with regard to association and classification rule mining on a variety of real datasets. Our experimental results indicate that, for a given privacy requirement, either substantially lower modeling errors are incurred as compared to the prior techniques, or the errors are comparable to those of direct mining on the true database.

An Optimal PMU Placement Method Against Measurement Loss and Branch Outage

This paper presents a new method for an optimal measurement placement of phasor measurement units (PMUs) for power system state estimation. The proposed method considers two types of contingency conditions (i.e., single measurement loss and single-branch outage) in order to obtain a reliable measurement system. First, the minimum condition number of the normalized measurement matrix is used as the criteria in conjunction with the sequential elimination approach to obtain a completely determined condition. Next, a sequential addition approach is used to search for necessary candidates for single measurement loss and single-branch outage conditions. These redundant measurements are optimized by binary integer programming. Finally, in order to minimize the number of PMU placement sites, a heuristic technique to rearrange measurement positions is also proposed. Numerical results on the IEEE test systems are demonstrated

Improving four-parameter sine wave fitting by normalization

Matrix equations need to be solved in each iterative round of four-parameter sine wave fitting. To ensure the robustness, normalizing factors rendering the minimum condition number (MCN) of the normal equation coefficient matrix was studied for the practical scenario (large sample size, and frequency well within [0, Nyquist Frequency]). Two parameter models, effective parameter model (EPM) and peak parameter model (PPM) were examined. The study shows that, firstly, with ignoring secondary entries of the coefficient matrix, the condition numbers depend solely on the amplitude of the four parameters. Secondly, for both models, there exists the optimal amplitude that renders the MCN. As amplitudes deviate from this optimal value, the condition number increases quickly. Thirdly, both the EPM and PPM possess a MCN close to 14. One numerical example shows that, if the realization data is normalized to the optimal amplitude, the condition numbers are indeed small, if the frequency lies in the practical range. With the initial frequency obtained from the method of the fast Fourier transform, outcomes at the fourth round iteration arrive to the Cramer-Rao bound.

A new aspect for choosing collocation points for solving biharmonic equations

This paper considers the Dirichlet problem for the biharmonic equation. Finite volume and multi-grid methods have been proposed for solving this equation but in this paper, a new idea will be investigated. This equation is reduced to a system of integral equations (SIEs). Collocation methods for solving some SIEs with weakly singular kernels are rather complicated so that a poor choice of a set of collocation points may lead to instability. Therefore, the best special grids highly concentrated near the possible singularity points for one of the kernels is investigated. Also, the SIEs is reduced to a convex functional, and a minimal solution of this functional gives the minimum condition number for the SIEs. A parameter of collocation points is defined such that it is related to minimal solution of the functional. On the other hand, the speed of convergence rate depends on selecting the parameter of collocation points and vice versa. Finally, we test our method using several numerical examples and demonstrate its efficiency.

Measurement placement for power system state estimation using decomposition technique

This paper applies the concept of distributed processing to the problem of measurement placement for power system state estimation. The proposed method uses the minimum condition number of the measurement matrix as a criterion in conjunction with sequential elimination to reach the near optimal placement positions. Firstly, the entire network of the power system is decomposed into smaller sub-networks. Then, in each sub-network, the optimal positions for measurement placement are determined by using the minimum condition number criteria. A heuristic algorithm for reducing the number of placement sites is also presented in order to minimize the communication costs. The numerical experimental results on the IEEE 14, 30, and 118 bus systems indicate that the proposed technique will provide a measurement matrix with smaller condition number and the computation time is much shorter than the non-decomposition approach.

An Optimal Measurement Placement Method for Power System Harmonic State Estimation

This paper focuses on a new technique for optimal measurement placement for power system harmonic state estimation (HSE). The solution provides the optimal number of measurements and the best positions to place them, in order to identify the locations and magnitudes of harmonic sources. The minimum condition number of the measurement matrix is used as the criteria in conjunction with sequential elimination to solve this problem. Two different test systems are provided to validate the measurement placement algorithm. A three-phase asymmetric power system has been tested using the New Zealand test system, while the IEEE 14-bus test system has been used for testing a balanced power system.

ROTATION METHOD FOR RECONSTRUCTING PROCESS AND FIELD FROM IMPERFECT DATA

Reconstruction of processes and fields from noisy data is to solve a set of linear algebraic equations. Three factors affect the accuracy of reconstruction: (a) a large condition number of the coefficient matrix, (b) high noise-to-signal ratio in the source term, and (c) no a priori knowledge of noise statistics. To improve reconstruction accuracy, the set of linear algebraic equations is transformed into a new set with minimum condition number and noise-to-signal ratio using the rotation matrix. The procedure does not require any knowledge of low-order statistics of noises. Several examples including highly distorted Lorenz attractor illustrate the benefit of using this procedure.

Getting exact information from the inverse jacobian matrix of parallel and serial robots

The inverse jacobian matrix of a robot contains highly useful information about the performances of the robot, for example about its accuracy and dexterity. Unfortunately these information are available only mostly through the eigenvalues of this matrix which are pose and design parameters dependent. We present interval analysis based algorithms that allow to extract useful information from any matrix such as minimal and maximal eigenvalues and minimal condition number.

Regularization of Discrete Ill-Posed Problems

The paper concerns conditioning aspects of finite-dimensional problems arising when the Tikhonov regularization is applied to discrete ill-posed problems. A relation between the regularization parameter and the sensitivity of the regularized solution is investigated. The main conclusion is that the condition number can be decreased only to the square root of that for the nonregularized problem. The convergence of solutions of regularized discrete problems to the exact generalized solution is analyzed just in the case when the regularization corresponds to the minimal condition number. The convergence theorem is proved under the assumption of the suitable relation between the discretization level and the data error. As an example the method of truncated singular value decomposition with regularization is considered.

An SVD-like matrix decomposition and its applications

A matrix S∈ C 2m×2m is symplectic if SJS ∗=J , where J= 0 I m −I m 0 . Symplectic matrices play an important role in the analysis and numerical solution of matrix problems involving the indefinite inner product x ∗( iJ)y . In this paper we provide several matrix factorizations related to symplectic matrices. We introduce a singular value-like decomposition B= QDS −1 for any real matrix B∈ R n×2m , where Q is real orthogonal, S is real symplectic, and D is permuted diagonal. We show the relation between this decomposition and the canonical form of real skew-symmetric matrices and a class of Hamiltonian matrices. We also show that if S is symplectic it has the structured singular value decomposition S=UDV ∗ , where U, V are unitary and symplectic, D= diag(Ω,Ω −1) and Ω is positive diagonal. We study the BJB T factorization of real skew-symmetric matrices. The BJB T factorization has the applications in solving the skew-symmetric systems of linear equations, and the eigenvalue problem for skew-symmetric/symmetric pencils. The BJB T factorization is not unique, and in numerical application one requires the factor B with small norm and condition number to improve the numerical stability. By employing the singular value-like decomposition and the singular value decomposition of symplectic matrices we give the general formula for B with minimal norm and condition number.

Rough set approach to incomplete information systems

In the paper we present Rough Set approach to reasoning in incomplete information systems. We propose reduction of knowledge that eliminates only that information, which is not essential from the point of view of classification or decision making. In our approach we make only one assumption about unknown values: the real value of a missing attribute is one from the attribute domain. However, we do not assume which one. We show how to find decision rules directly from such an incomplete decision table, which are as little non-deterministic as possible and have minimal number of conditions.

Twin-head six-axis force sensors

Three-axis force sensors based on the principle of a pressure pick-up device are characterized by their manufacturability, simple structure, and small size. Their use, however, has not been explored due to the incomplete force information that they provide. This paper proposes a new six-axis force sensor, composed of two three-axis force sensors and a connector, called a twin-head type six-axis force sensor (TH force sensor for short). The author first describes the theoretical basis for the operation of TH force sensors and provides the characteristic matrix connecting the load and sensor output vectors. While twenty-one types of TH force sensor are theoretically conceivable, the author proves that ten types can result in a characteristic matrix with a full rank. The author also considers an optimal design in the sense of minimum condition number.

A Trace Inequality for Unitary Matrices

(1994). A Trace Inequality for Unitary Matrices. The American Mathematical Monthly: Vol. 101, No. 5, pp. 453-455.

Prediction of voltage collapse in an integrated AC-DC network using the singular-value decomposition concept

Abstract A power system under heavy loading conditions may face the problem of voltage instability followed by voltage collapse. The minimum singular value of the AC power flow Jacobian has been popularly used as a static voltage stability index showing the distance between the steady-state operating point and the point of voltage collapse. In this paper, the concept of minimum singular value and condition number of the load flow Jacobian has been extended to study the voltage instability of a combined AC-DC system. The proposed method has been applied to determine the voltage collapse point in four sample AC-DC systems, three of them derived from the IEEE 14-bus system and the fourth a practical 156-bus Indian system. The model includes a switching scheme to check the converter control parameter limit violations. The effect of load changes at a single bus as well as at all the buses simultaneously on voltage collapse has also been studied.

Structures of p-isometric matrices and rectangular matrices with minimum p-norm condition number

This paper discusses the structure of rectangular matrices with minimum p-norm condition number and the structure of p-norm isometric matrices (1 ⩽ p ⩽ +∞). Bounds on the condition number of an isometric matrix are also given.

Some computational issues in optimal control by nonlinear programming

The Roppenecker [11] parameterization of multi-input eigenvalue assignment, which allows for common open- and closed-loop eigenvalues, provides a platform for the investigation of several issues of current interest in robust control. Based on this parameterization, a numerical optimization method for designing a constant gain feedback matrix which assigns the closed-loop eigenvalues to desired locations such that these eigenvalues have low sensitivity to variations in the open-loop state space model was presented in Owens and O'Reilly [8]. In the present paper, two closely related numerical optimization methods are presented. The methods utilize standard (NAG library) unconstrained optimization routines. The first is for designing a minimum gain state feedback matrix which assigns the closed-loop eigenvalues to desired locations, where the measure of gain taken is the Frobenius norm. The second is for designing a state feedback matrix which results in the closed-loop system state matrix having minimum condition number. These algorithms have been shown to give results which are comparable to other available algorithms of far greater conceptual complexity.

Kinematic Isotropy and the Conditioning Index of Serial Robotic Manipulators

The conditioning index of a serial robotic manipulator is de fined in this article in terms of the reciprocal of its minimum condition number. The condition number of a manipulator is defined, in turn, as that of its Jacobian matrix. Moreover, in defining the Jacobian condition number, a quadratic norm of the Jacobian matrix is needed. However, this norm, or for that matter any other norm, brings about dimensional inho mogeneities. It is shown here that by properly defining the said norm based on a weighting positive definite matrix, the dimensional inhomogeneity is resolved. Manipulators with a conditioning index of 100% are termed isotropic, a six-axis isotropic manipulator being introduced. This manipulator has all its angles between neighboring revolute axes at 90° and all its distances between neighboring axes identical; more over, these distances are identical to the offsets of those axes. The kinematic conditioning of wrist-partitioned manipulators is given due attention, and illustrated with some examples of industrial robots of this type.

Is the Relative Gain Array a Sensitivity Measure?

Abstract Whether the relative gain array (RGA) can be used to reliably predict sensitivity with respect to uncertainty is of significant practical importance for resilient process design and process control. The theory basis seems to link the RGA to the minimal Euclidean condition number (MECN). This paper will derive an equality relation for 2×2 and 3×3 processes and an inequality for systems with dimension 4. The conjecture of Nctt and Manousiouthakis (1987) is theoretically clarified, and a numerical example is constructed to check our results.