Eigenvector Assignment Research Articles

The Adjugate Method: Reassignment with System Classification

Further elaborations on the adjugate method for eigenvalue–eigenvector assignment are discussed in this paper. Additional results concerning the admissible pair introduced recently are stated, verified, and commented on. Properties of the adjugate method concerning eigenvalue reassignment of controllable and uncontrollable systems are revealed and pursued further in this study A reformulation of how calculations should be carried out is investigated, pointing out merits, bounds, and limitations. It has been found that reassignment can involve a case where the closed-loop eigenvector companion zi is zi=0 and the associated eligible closed-loop eigenvectors wi are open-loop ones. This can be considered an advantage in the sense of knowing zi and wi beforehand. This also applies to the case of uncontrollable eigenvalue reassignment, where enlarged closed-loop eigenvector subspaces are uncovered, enabling more flexible designs. Assessments of the traditional method compared to the adjugate method are commented on whenever appropriate. Numerical issues are pointed out where Leverrier’s algorithm is adapted for efficient matrix null space and matrix adjugate determination. A feature of the adjugate method is a result where wU=0, indicating that the eigenvalue assigned is uncontrollable. Such a distinctive feature enables a classification procedure for systems regarding controllability and observability. The various concepts pointed out have been demonstrated and authenticated through carefully selected examples.

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick’s constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples.

Multistage output-lifting eigenstructure assignment: a multirate ball and plate example

ABSTRACTBy exploiting both the left and the right allowable subspaces in consecutive stages, this paper extends a recently developed output-lifting eigenstructure assignment approach into a multistage eigenstructure assignment scheme. In this scheme, design degrees of freedom, enlarged via output-lifting, are further exploited to improve eigenvector assignment. To mitigate the inherent conflicts between the theoretical development of eigenstructure assignment and inherent physical system characteristics, the paper also clearly demonstrates how to derive an ideal eigenstructure, particularly the desired eigenvectors, to distribute and decouple the natural modes among appropriate states or outputs, based upon an example: a novel multirate ball and plate system. The design and simulation results show the efficacy of the scheme.

Eigenstructure Assignment for Componentwise Ultimate Bound Minimization in Discrete-Time Linear Systems

This technical note presents a feedback design approach to minimise state ultimate-bounds (UBs) in linear time-invariant discrete-time systems subject to non-vanishing bounded disturbances. A target closed-loop eigenstructure is first constructed to achieve the lowest possible UBs for one or more state components of the system. Structural conditions on the system matrices are then derived to guarantee that the target eigenstructure can be assigned by state feedback. When the required structural conditions are satisfied, an eigenvector assignment procedure can be applied to simultaneously minimise multiple UBs in systems with multiple inputs. The results are illustrated by a numerical example drawn from a distributed voltage control problem in inverter-based microgrids.

Improving active eigenvector assignment through passive modifications

Specifications on the dynamic behavior of feedback-controlled vibrating systems are often expressed in terms of its eigenstructure, i.e. eigenvalues and eigenvectors. The notion of controllability establishes the possibility to assign eigenvalues through state feedback, but it is not adequate to assure the assignment of arbitrary eigenvectors. Indeed, assignable eigenvectors are just those belonging to the allowable vector subspace, which depends on the physical properties of the vibrating system (mass, damping and stiffness matrices) and of the actuators. To overcome this limitation, this paper proposes a hybrid approach that exploits passive modification of the system physical parameters to modify the allowable subspace in such a way that it spans (or closely approximates) the desired eigenvectors. Then, once that the system modifications have been computed, standard techniques for control synthesis can be employed to compute the gains assigning the desired poles and the eigenvectors. The modification of the allowable subspace is cast in this work as a rank minimization problem, which can be efficiently tackled through semi-definite programming. The proposed method is numerically validated on a lumped parameter system, by proving that the assignment of eigenvectors by hybrid control is significantly enhanced compared with sole active control.

On Stabilization of Switched Linear MI Systems Using in Part Common Left Eigenvector Assignment Based on LMIs

The problem of assigning in part common left eigenvector assignment by the state feedback control of linear multi-input systems is considered in the paper. Based on linear matrix inequalities characterization, the exposition of the problem is formulated to handle a common left eigenvector for switched modes. Limited by preexisting degrees of freedom, which factually makes mutually contingent subsets of the closed-loop eigenvalue spectrum of each mode, the design conditions for in part common left eigenvector assignment of all modes are derived in the terms of linear matrix inequalities. The procedure provides a way available in linear switched control system design while the state-space control structures take standard forms. An example is included to assess the properties of the technique and its application.

Fast‐rate residual generator based on multiple slow‐rate sensors

This study puts forward the problem of fast-rate fault detection based on multiple slow-rate sensors. A fast-rate residual generator with casuality constraint is established from the multi-sensor model. Parameters of the residual generator are determined via disturbance-decoupling based on left eigenvector assignment. It is found that the condition of disturbance-decoupling is related to the multi-rate sensor sampling nature. A numerical example is given to illustrate the effectiveness of the proposed residual generator.

Study on Class of Automated Guided Vehicle Stability Based on Dynamic Output Observer

In automated guided vehicle(AGV) system, a stability method, which is based on dynamic output observer, is put forward to solve the stability problem of trajectory tracking system. The model of AGV trajectory tracking system is built. The shortage of design method of dynamic output observer is discussed. It is proved that the condition number of eigenvector matrix has important influence on the system's robust stability. Because of the freedom of eigenvector assignment, the particle swarm arithmetic is used to optimize dynamic output observer. The freedom of eigenvector assignment is represented by the particle's position, and the condition number of eigenvector matrix is used to be the objective function. The particle swarm arithmetic is used to obtain the minimum value of the condition number. It is shown by simulation that the robust stability of dynamic output observer with particle swarm arithmetic is better than that of dynamic output observer and full-order observer.

Dependent modal space control

This paper presents a new control technique for reducing vibration in flexible structures. It is based on the modal approach and is called dependent modal space control (DMSC). The well-known independent modal space control (IMSC) method, devised in the 1980s, allows the frequency and damping of the controlled modes to be changed using diagonal control gain matrices, leaving the mode shapes unaltered. DMSC, on the other hand, can impose not only frequency and damping but also the controlled mode shapes by using full control gain matrices. In many applications, owing to the limited number of sensors–actuators available for control and the increasing spillover effects, generic assignment of eigenvectors is not possible. However, an optimal eigenstructure assignment can be computed to reduce structure vibration by minimizing an input–output performance index. To demonstrate the advantages of this new method, we compare IMSC and DMSC using numerical simulation on a finite element method model of a cantilevered beam.

Feedback Stabilization of Switching Discrete-Time Systems via Lie-Algebraic Techniques

This technical note addresses the stabilization of switching discrete-time linear systems with control inputs under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems, which is ensured when the closed-loop component subsystem matrices are stable and generate a solvable Lie algebra. The present work develops an iterative algorithm that seeks the feedback maps required for stabilization based on the previous Lie-algebraic condition. The main theoretical contribution of the technical note is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution. The core of the proposed algorithm is a common eigenvector assignment procedure, which is executed at every iteration. We also show how the latter procedure can be numerically implemented and provide a key structural condition which, if satisfied, greatly simplifies the required computations.

Control of static synchronous compensator with supplementary damping enhancement for wind farm voltage regulation

This study presents a new control design approach for the wind farm voltage regulation using static synchronous compensator (STATCOM). A mode decoupling proportional plus integral (PI) controller, in which the control gains are synthesised by eigenstructure assignment (EA), is proposed for the STATCOM. EA is a state-space approach, which is used to improve system damping by eigenvalue assignment and alter system mode shape by eigenvector assignment. To embed the physical PI control loops in the state-space model, an equivalent transformation is proposed to map the feedback gain matrix to the PI controller. Owing to the presence of the physical PI control loops, the control signals would not mire into saturation when appropriate limiter according to the STATCOM capacity is placed in the control variables. To further alleviate the undesired oscillation mode excited by transient disturbance, a lead-lag-based supplementary damping controller (SDC) is adopted. Simulation comparisons are provided to show that the proposed technique can be an effective method for mitigating the voltage fluctuation in the wind farm because of the load changes and the wind gust.

Reduction of Structural Acoustic Radiation Via Left and Right Eigenvector Assignment Approach

The objective of this research is to investigate the feasibility of utilizing a new left—right eigenvector tailoring method in reducing the acoustical radiations of flexible structures. The structural sound pressure radiation can be expressed in terms of a combination of vibration modes, where its magnitude is also a function of the external disturbance distribution. In other words, the radiated sound pressure level depends on both the right eigenvectors (related to the structural mode shapes) and left eigenvectors (related to the system disturbance rejection ability) of the vibrating structure. The basic idea of the proposed approach is to simultaneously modify the structural modal velocity distribution and the system capability of disturbance rejection through active left—right eigenvector assignment control actions, so that the sound pressure radiated from the vibrating structure can be reduced. Numerical simulations are performed to evaluate the effectiveness of the proposed method on structural noise reduction. Frequency responses of sound pressure at a receiver in the selected frequency range are illustrated. It is shown that with the proposed active control method, one can re-shape the modal velocity distribution and enhance disturbance rejection, and hence can effectively minimize the structural sound pressure radiation.

An adaptive left–right eigenvector evolution algorithm for vibration isolationcontrol

The purpose of this research is to investigate the feasibility of utilizing an adaptive left andright eigenvector evolution (ALREE) algorithm for active vibration isolation. As depictedin the previous paper presented by Wu and Wang (2008 Smart Mater. Struct. 17015048), the structural vibration behavior depends on both the disturbance rejectioncapability and mode shape distributions, which correspond to the left and righteigenvector distributions of the system, respectively. In this paper, a novel adaptiveevolution algorithm is developed for finding the optimal combination of left–righteigenvectors of the vibration isolator, which is an improvement over the simultaneousleft–right eigenvector assignment (SLREA) method proposed by Wu and Wang(2008 Smart Mater. Struct. 17 015048). The isolation performance index used inthe proposed algorithm is defined by combining the orthogonality index of lefteigenvectors and the modal energy ratio index of right eigenvectors. Through theproposed ALREE algorithm, both the left and right eigenvectors evolve such that theisolation performance index decreases, and therefore one can find the optimalcombination of left–right eigenvectors of the closed-loop system for vibration isolationpurposes. The optimal combination of left–right eigenvectors is then synthesizedto determine the feedback gain matrix of the closed-loop system. The result ofthe active isolation control shows that the proposed method can be utilized toimprove the vibration isolation performance compared with the previous approaches.

An Eigenstructure Assignment for a Static Synchronous Compensator

Problem statement: Power flow through an AC transmission line is influenced by three basic electrical parameters, which are line impedance, magnitudes and phase-shift angle between the sending and receiving voltages. Therefore, the change in any of the three basic parameters means a change in the power flow through the transmission line. The aims of this research paper are: increase the power transfer capability of transmission systems, minimize the transmission losses, support a good voltage profile and retain system stability under large disturbances. Study the use of eigenstructure techniques for state feedback control of the power system static compensator. Therefore, the mathematical analysis was performed for eigenvector assignment, power flow transmission line and for the static compensator analysis based on the transformation of the three-phase into d-q frame. Approach: A novel control method for regulating the power system in case of abnormal conditions was carried out. The system considered is a static synchronous compensator. The study includes a detailed mathematical analysis of the impact of the shunt compensator on the power flow; investigation of the system constraints and their effects on the static compensator control; in addition simulation of static compensator to control a transmitted active power flow on the transmission line. The conducted method provides a way of constructing the state feedback gain matrix to satisfy a certain prescribed performance. Results: The solutions of the obtained equation were conducted using the computer simulation method for both open-loop and static compensator techniques. The result shows fast tracking of the power flow transient response when using the static compensator technique comparing with open-loop technique. However, the same trend of the behavior was observed for all cases. Conclusion: A new method for developing a parameterized feedback matrix that assigns a closed-loop prespecified set of eigenvalues was obtained. It improves the overall system performance and yields a class of controllers contributing uniformly to the assignment process. The voltage could be kept constant independent of the loads with static compensator. The results show clearly the applicability of the proposed control scheme which is acceptable for the static compensator.

Disturbance Attenuation in Fault Detection of Gas Turbine Engines: A Discrete Robust Observer Design

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This study is motivated by the onboard fault detection of gas turbine engines (GTEs), where the computation resources are limited and the disturbance is assumed to be band-limited. A fast Fourier transformation (FFT)-based disturbance frequency estimation approach is proposed and performance indexes are improved by integrating such frequency information. Furthermore, in the left eigenvector assignment, both eigenvalues and free parameters are optimized. As illustrated in the application to the actuator fault detection of a GTE, significant improvements are achieved compared to the existing methods. By combining the frequency estimation and eigenvalue optimization, the main contribution of the paper is the reduction of the computation complexity and the avoidance of the local optimal solution due to fixed eigenvalues. </para>

Active vibration isolation via simultaneous left–right eigenvector assignment

The objective of this research is to synthesize a simultaneous left and right eigenvectorassignment (SLREA) method for active vibration isolation. It is a pioneering effort toutilize such an eigenvector assignment concept for active isolator design, where theapproach can provide good physical insight into the problem. In this investigation, a newalgorithm for the synthesis of the desired left eigenvectors is developed, which is animprovement over the classical methods. The purpose of the right eigenvectorassignment method is to alter the closed-loop system modes such that the modalcomponents corresponding to the concerned region (isolation area of the isolator) haverelatively small vibration amplitude. Correspondingly, the design goal of the lefteigenvector assignment is to alter the left eigenvectors of the closed-loop system sothat they are as closely orthogonal to the system’s forcing vectors as possible.With the proposed approach, one can achieve both disturbance rejection andmodal confinement concurrently for the purpose of vibration isolation. In thisresearch, a new formulation is developed so that the desired left eigenvectorsof this integrated system are selected through solving a generalized eigenvalueproblem, where the orthogonality indices between the forcing vectors and the lefteigenvectors are minimized. The components of the right eigenvectors correspondingto the concerned region are minimized concurrently. It is shown that, with theSLREA technique, both disturbance rejection and modal confinement can beachieved, and thus vibration amplitude in the isolated region can be suppressedsignificantly.

Periodic Isolator Design Enhancement via Vibration Confinement through Eigenvector Assignment and Piezoelectric Circuitry

The objective of this research is to investigate the feasibility of utilizing eigenvector assignment and piezoelectric circuitry for enhancing vibration isolation performance of periodic isolators. For a classical periodic structure, stop bands are created due to material discontinuity so that wave propagation of external excitation can be suppressed within the stop band frequency range. While effective, such a method cannot always create wide enough stop bands such that all disturbance frequencies are covered. In this study, the eigenvector assignment technique and piezoelectric circuitry are utilized to reduce the transmissibility of the isolator modes near the boundary of the stop bands, and therefore widen the effective frequency range of vibration suppression of the periodic isolator. The principle of eigenvector assignment is to alter the mode shapes of the system so that the modal components corresponding to the concerned coordinates are as small as possible. By applying the eigenvector assignment method on the spatially tailored periodic isolator structure, the response amplitude of the attenuated end (the end of the isolator designed to have small vibration) at resonant frequencies near the stop band can be reduced, which enhances the vibration isolation performance in the frequency range of interest. On the other hand, piezoelectric circuits connecting to the isolator structure increase the degrees of freedom of the integrated system, and enlarge the design space for achievable eigenvectors. The eigenvectors of this integrated system are selected such that the modal energy in the concerned coordinates is minimized by using the Rayleigh Principle. The integrated system with assigned eigenvectors will re-distribute vibratory energy of the complete electromechanical system. Small vibration at the attenuated end of the isolator is achieved since the energy is confined in the circuitry and other parts of the isolator. Numerical simulations are performed to evaluate the effectiveness of the proposed method on vibration confinement for isolator designs. An integrated closed-loop system with state estimator is developed to realistically implement the proposed algorithm. It is shown that with the piezoelectric circuitry and eigenvector assignment, the system energy is redistributed and confined in the unconcerned regions, which can greatly enhance the performance of the vibration isolation system.

System output feedback controller design via O.S.A. method

The paper addresses the design of a system stabilizer using an optimal reduced order model by only the output state variables. System damping can be improved by using eigenvalues assignment and the co-ordination of stabilizers can be achieved through eigenvector assignment by maintaining system mode shape, this is defined as the optimal subeigenstructure assignment (O.S.A) method. The proportional-integral controller is derived via the optimal reduced order model instead of via the whole system model. The effectiveness of this stabilizer is evaluated, and this study reveals that the result of eigenstructure assignment is more stable and much better than in the assignment method based on the whole system model. A system is given as examples to illustrate the advantages and effectiveness of the proposed approach. Results based on the whole system model are included for comparison.

Mistuning Identification and Model Updating of an Industrial Blisk

The results of a complete study of mistuning identification on an industrial blisk are presented. The identification method used here is based on a model-updating technique of a reduced order. This reduced-order model is built using component mode synthesis, and mistuning is introduced as perturbations of the cantilevered-blade modes. The measured modal data are extracted from global measurements of the blisk's forced response. As we use a single point excitation, this measurement procedure allows the acquisition of all the modes of a given family with a quite simple experimental set-up. A selection of the best identified modal data is finally performed. During the mistuning identification procedure, these measured data are regularized using an eigenvector assignment technique which reduces the influence of eventual measurement errors. An inverse problem, based on the perturbed (mistuned) modal equation, is defined with measured modes as input and mistuning parameters as unknown. Then, the reduced-order model is updated with the identified mistuning, we first perform a correlation on modal responses (using eigenfrequency deviation criteria and MACs). Finally, correlation results on forced responses are presented and discussed.

Magnetic excitations and phonons in the spin-chain compoundNaCu2O2

We report an inelastic light scattering study of single-crystalline NaCu$_2$O$_2$, a spin-chain compound known to exhibit a phase with helical magnetic order at low temperatures. Phonon excitations were studied as a function of temperature and light polarization, and the phonon frequencies are compared to the results of ab-initio lattice dynamical calculations, which are also reported here. The good agreement between the observed and calculated modes allows an assignment of the phonon eigenvectors. Two distinct high-energy two-magnon features as well as a sharp low-energy one-magnon peak were also observed. These features are discussed in terms of the magnon modes expected in a helically ordered state. Their polarization dependence provides evidence of substantial exchange interactions between two closely spaced spin chains within a unit cell. At high temperatures, the spectral features attributable to magnetic excitations are replaced by a broad, quasielastic mode due to overdamped spin excitations.