On the analysis of eigenvalue assignment robustness

Algorithms are currently available for the solution of certain types of optimal-eigenvalue-assignment problems in which the eigenvalues of a given system are required to be shifted into preassigned locations or region while also minimising an appropriate quadratic-performance criterion. All the known methods for a solution of the above problem are based on manipulation of the original nth-order system matrices even if only r eigenvalues (r ≥ n) of the original system are to be reassigned. On the contrary, the method proposed in this paper, for a solution of the above problem, employs an rth-order equivalent model, which leads to a solution via manipulation of rth-order matrices only. The method also ensures that the remaining n — r eigenvalues of the original system are not disturbed and are carried over to the resultant feedback system. It has been shown in the paper that the suggested procedure brings about a considerable saving in computation time, and also requires less computer storage. Two numerical examples have been included to illustrate the suggested technique.

For a single-output linear time-invariant system, output controllability is shown to be a necessary and sufficient condition for the existence of linear state variable feedback that freely assigns the eigenvalues of the closed-loop system that are observable from the output. An application of this result to the single-output linear servomechanism shows that the ability of such a system to track with zero steady-state error and an arbitrary transient response is equivalent to output controllability of the plant, and is independent of the class of command signals and disturbances.

The partial eigenvalue assignment problem concerns reassigning afew of undesired eigenvalues of a linear system to suitably chosenlocations and keeping the other large number of eigenvalues andeigenvectors unchanged (no spill-over). This paper considers thepartial eigenvalue assignment problem with time delay robustness.A time delay robustness measure is presented by analyzing thesensitivity of the assigned eigenvalues with respect to timedelay. The problem is formulated as an unconstrained minimizationproblem with the cost function involving the time delay robustnessmeasure. A numerical algorithm with analytical formulation of thegradient for the cost function is provided. A numerical example isincluded to show the effectiveness of the proposed method.

It is shown that in uncontrollable linear system ẋ = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix  M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A  − BK has the desired eigenvalues. The procedure is illustrated by simple example.

In this paper, we present some results on robustness of eigenvalue distribution in specified complementary regions for perturbed systems. If some eigenvalues of the nominal system are located in a specified region, the proposed sufficient conditions guarantee that the same number of eigenvalues of the perturbed system lie inside the same region. The characteristics of a linear time-invariant system are influenced by the eigenvalue location of the system matrix. Due to uncertainty or parameter variation, all mathematical descriptions of dynamic systems are approximate models at best. The effect of uncertainty will move the eigenvalues of a real system away from the designed ones. Therefore, it is significant to guarantee that the same number of eigenvalues of the perturbed system lie inside the same region as that of the nominal system. By the analysis of eigenvalue distribution, we can explore the locations of dominant eigenvalues, specified eigenvalues or even individual eigenvalues of perturbed systems. Consequently, more properties of perturbed systems such as stability margin, performance robustness and so on can be examined. The proposed theorems can be applied to both continuous- and discrete-time systems. In addition, the analysis of stability robustness can be dealt with as a special case in our study. Two examples are given to show the applicability of the proposed theorems. Finally, some conclusions are presented.

The authors examine the problem of robust pole placement using state-feedback in generalized systems. They develop a robustness theory for the finite generalized spectrum of the system as a partial problem. The basic tool that is exploited in this theory is the concept of chordal metric. The feedback laws presented always guarantee the closed-loop regularity. These results led to necessary and sufficient conditions for perfect conditioning, and showed how the results for optimal conditioning can be used for the generalized case. >

This paper concerns the state estimation problem for linear discrete-time systems with non-Gaussian state and output noises. A sub-optimal quadratic filter algorithm is proposed. In order to enlarge the class of systems allowed to be processed, a novel approach based on the output injection stabilization is derived. Also a second benefit to the estimate performances, due to the possibility of assignment of the system eigenvalues, is investigated. Numerical results validate the effectiveness of the proposed method.

This work studies the problem of design of nonlinear observers in the presence of exogenous disturbances. In particular, the present work proposes a systematic design method for nonlinear state and disturbance observers in modular form, consisting of an observer for the disturbance-free part of the system, along with a disturbance observer and a state-estimate corrector. The modular observer is first defined and characterized in a general setting and then, a systematic design method is developed on the basis of exact linearization with eigenvalue assignment. Necessary and sufficient conditions for feasibility of the proposed nonlinear modular observer design are derived. The proposed approach is a direct generalization of the modular linear Luenberger observer to nonlinear systems

This study presents a novel reduced-order method to find critical eigenvalues of ultra large-scale power system. First, the numerical solution of matrix exponential is computed by precise time-step integration. Secondly, based on the numerical solution, the numerical curve of the trace of matrix exponential is formed. Thirdly, the candidates of critical eigenvalues are extracted from the numerical curve of the trace by Prony analysis, and finally, a set of weight coefficients is calculated to confirm critical eigenvalues from candidate eigenvalues. Since the trace contains all eigenvalues, no critical eigenvalue can be lost in analysing the numerical curve of the trace. In the later period of time integration, the effect of all non-critical eigenvalues to the trace is decayed and oppositely the effect of all critical eigenvalues is amplified. Thus, several rightmost eigenvalues can only be extracted from the numerical curve of the trace. Case studies for 16 and 9004 order system have validated that the proposed method can be used to find all critical eigenvalues of ultra large-scale power system.

A simple and efficient approach is needed for robustness evaluation and optimization of treatment planning in routine clinical particle therapy. Here we propose a robustness analysis method using dose standard deviation (SD) in possible scenarios such as the robustness indicator and a fast dose warping method, i.e. deformation of dose distributions, taking into account the setup and range errors in carbon-ion therapy. The dose warping method is based on the nominal dose distribution and the water-equivalent path length obtained from planning computed tomography data with a clinically commissioned treatment planning system (TPS). We compared, in a limited number of scenarios at the extreme boundaries of the assumed error, the dose SD distributions obtained by the warping method with those obtained using the TPS dose recalculations. The accuracy of the warping method was examined by the standard-deviation-volume histograms (SDVHs) for varying degrees of setup and range errors for three different tumor sites. Furthermore, the influence of dose fractionation on the combined dose uncertainty, taking into consideration the correlation of setup and range errors between fractions, was evaluated with simple equations using the SDVHs and the mean value of SDs in the defined volume of interest. The results of the proposed method agreed well with those obtained with the dose recalculations in these comparisons, and the effectiveness of dose SD evaluations at the extreme boundaries of given errors was confirmed from the responsivity and DVH analysis of relative SD values for each error. The combined dose uncertainties depended heavily on the number of fractions, assumed errors and tumor sites. The typical computation time of the warping method is approximately 60 times less than that of the full dose calculation method using the TPS. The dose SD distributions and SDVHs with the fractionation effect will be useful indicators for robustness analysis in treatment planning, and the results of our comparative study show that the proposed analysis method would be beneficial in routine clinical use.

The use of active vibration control technique, though typically more expensive and difficult to implement compared to the traditionally used passive control approaches, is an effective way to control or suppress dangerous vibrations in structures caused by resonance or flutter due to the external disturbances, such as winds, waves, moving weights of human bodies, and earthquakes. From control perspective, it is not only sufficient to suppress the resonant modes of vibration but also to guarantee the overall stability of the structural system. In addition, the no-spillover property of the remaining large number of frequencies and the corresponding mode shapes must be maintained. To this end, the state-space based linear quadratic regulator (LQR) and \(H_{\infty }\) control techniques and physical-space based robust and minimum norm quadratic partial eigenvalue assignment (RQPEVA and MNQPEVA) techniques have been developed in the recent years. In contrast with the LQR and \(H_{\infty }\) control techniques, the RQPEVA and MNQPEVA techniques work exclusively on the second-order model obtained by applying the finite element method on the vibrating structure, and therefore can computationally exploit the nice properties, such as symmetry, sparsity, bandness, positive definiteness, etc., inherited by the system matrices associated with the second-order model. Furthermore, the RQPEVA and MNQPEVA techniques guarantee the no-spillover property using a solid mathematical theory. The MNQPEVA technique was originally developed for the deterministic models only, but since then, further extension to this technique has been made to rigorously account for parametric uncertainty. The stochastic MNQPEVA technique is capable of efficiently estimating the probability of failure with high accuracy to ensure the desired resilience level of the designed system. In this work, first a brief review of the LQR, \(H_{\infty }\), and MNQPEVA techniques is presented emphasizing on their relative computational advantages and drawbacks, followed by a comparative study with respect to computational efficiency, effectiveness, and applicability of the three control techniques on practical structures. Finally, the methodology of the stochastic MNQPEVA technique is presented and illustrated in this work using a numerical example with a representative real-life structure and recorded data from a California earthquake.

The present paper proposes an in situ modal parameter-based method for determining the dynamic joint characteristics of mechanical systems. The proposed method uses the mass, damping and stiffness matrices of the structure, calculated by the finite element method (FEM), along with measured eigenvectors and eigenvalues of the actual system. While the modal parameters at the joint must be known in order to identify the structural joint dynamic characteristics, it is often impossible to measure the response of the system directly at the joint location. To overcome this problem and eliminate the errors associated with using measurements close to the joints, an alternative indirect estimation scheme is used to determine the complete set of eigenvectors, and thus the eigenvector component corresponding to the joint location is extracted. Therefore, the proposed method allows in situ joint parameter identification. The efficiency of this method is validated by simulations with different mechanical systems, and the robustness is also demonstrated with errors introduced into the estimated eigenvectors. Finally, the method is implemented for experimental identification of the joint parameters of an actual spindle system.

Dynamic magnetic resonance imaging (dMRI) strikes a balance between reconstruction speed and image accuracy in medical imaging field. In this paper, an improved robust tensor principal component analysis (RTPCA) method is proposed to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data. The MR reconstruction problem is formulated as a high-order low-rank tenor plus sparse tensor recovery problem, which is solved by robust tensor principal component analysis (RTPCA) with a new tensor nuclear norm (TNN). To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into TNN for enforcing the low-rank structure in MRI datasets. The experimental results show that the proposed method outperforms state-of-the-art methods in terms of both MR image reconstruction accuracy and computational efficiency on 3D and 4D experiment datasets, especially for 4D MR image reconstruction. Graphical abstract The flowchart of the proposed method to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data in the kth iteration. To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into tensor nuclear norm (TNN) for enforcing the low-rank structure in MRI datasets. In each iteration, the first step is to get low-rank tensor ℓk - 1 by using soft thresholding on the singular values of ℓk - 1 = χk - 1 - ξk - 1, and an improved tensor nuclear norm method is proposed to process the low-rank tensor ℓk - 1 firstly. Then, the shrinkage operator is applied to ξk - 1 = χk - 1 - ℓk - 1 for sparse part ξk - 1. The final reconstructed d-MRI χk is obtained by enforcing data consistency that the residual in K-space is subtracted by the sum of the reconstructed low-rank tensor and sparse tensor.

The reliability-based analysis of frequency of the vibration control problem of structures with random parameters is discussed. Based on the theory of perturbation method, closed-loop vibration control systems, reliability analysis, and PNET method, the method of reliability-based analysis of eigenvalues of closed-loop vibration control systems with uncertain parameters is studied. And the distribution function of the random eigenvalues will not be computed other than their means and variances. The standard deviations of eigenvalues of the uncertain closed-loop systems can be used to estimate the reliability of frequency. The numerical results show that the present method is effective.

This paper presents a novel method for handling the effects of shape outliers in statistical shape analysis. Usually performed by a variant of classical principal component analysis (PCA), variability analysis may be highly affected by erroneous shapes. Principal components may thus imply aberrant modes, while eigenshapes may not accurately describe variability in a given set of shapes. Our robust analysis is performed using an elastic metric associated with the square-root velocity representation of shapes. This elastic shape analysis allows shape variability to be described with natural and intuitive deformations. The proposed method based on shape outlier detection applies the shape restoration procedure to rectify aberrant shapes. The resultant components are thus obtained from a tangent PCA on the restored database. By performing experiments based on MPEG-7 and HAND databases, we demonstrate that the proposed scheme is effective for shape variability analysis in the presence of outlying shapes. Our method is then compared with two existing schemes for robust data variability analysis: minimum covariance determinant-based PCA and projection pursuit-based PCA.