System Dynamic Matrix Research Articles

Remeshing‐free Graph‐based Finite Element Method for Fracture Simulation

AbstractFracture produces new mesh fragments that introduce additional degrees of freedom in the system dynamics. Existing finite element method (FEM) based solutions suffer from increasing computational cost as the system matrix size increases. We solve this problem by presenting a graph‐based FEM model for fracture simulation that is remeshing‐free and easily scales to high‐resolution meshes. Our algorithm models fracture on the graph induced in a volumetric mesh with tetrahedral elements. We relabel the edges of the graph using a computed damage variable to initialize and propagate fracture. We prove that non‐linear, hyper‐elastic strain energy density is expressible entirely in terms of the edge lengths of the induced graph. This allows us to reformulate the system dynamics for the relabelled graph without changing the size of the system dynamics matrix and thus prevents the computational cost from blowing up. The fractured surface has to be reconstructed explicitly only for visualization purposes. We simulate standard laboratory experiments from structural mechanics and compare the results with corresponding real‐world experiments. We fracture objects made of a variety of brittle and ductile materials, and show that our technique offers stability and speed that is unmatched in current literature.

Robust static output feedback control for hidden Markov jump linear systems

This paper studies the robust static output feedback control of discrete-time Markov jump systems considering that the mode of operation cannot be directly measured. It is assumed that the only available information concerning the main jump process comes from a detector, so that the jump and detector processes can be modelled as a hidden Markov model. Initially, it is considered that the system's dynamic matrix is subject to parametric uncertainties. By assuming a full row rank condition for the output matrix and employing a clusterisation technique for the state space of the hidden Markov model, it is derived a set of linear matrix inequalities (LMIs) that guarantees robust stability in the mean-square sense of the closed-loop system, as well as a bound on its norm. It is also shown that this design technique is linked to the bounded real lemma, so that a new set of LMIs conditions can be obtained for the ‘pure’ control problem, which can be extended to cover the robust polytopic case. The paper is concluded with an illustrative example.

Tensor decomposition for multi-agent predictive state representation

Predictive state representation (PSR) uses a vector of action-observation sequence to represent the system dynamics and subsequently predicts the probability of future events. It is a concise knowledge representation that is well studied in a single-agent planning problem domain. To the best of our knowledge, there is no existing work on using PSR to solve multi-agent planning problems. Learning a multi-agent PSR model is quite difficult especially with the increasing number of agents, not to mention the complexity of a problem domain. In this paper, we resort to tensor techniques to tackle the challenging task of multi-agent PSR model development problems. By first focusing on a two-agent setting, we construct the system dynamics matrix as a high order tensor for a PSR model, learn the prediction parameters and deduce state vectors directly through two different tensor decomposition methods respectively, and derive the transition parameters via linear regression. Subsequently we generalize the PSR learning approaches in a multi-agent setting. Experimental results show that our methods can effectively solve multi-agent PSR modelling problems in multiple problem domains.

Dynamic crack face contact and propagation simulation based on the scaled boundary finite element method

This paper extends the static contact problem of crack faces to include dynamic contact and crack propagation based on the relevant literature (Zhang et al., 2018). The scaled boundary finite element method is used to solve dynamic crack face contact and crack propagation problems. Based on the scaled boundary finite element shape function considering side-face loads, the system dynamic equilibrium equation and related characteristic matrix are presented. The Lagrange multiplier method is used to establish the contact model between the crack faces. The meshing and remeshing algorithm is carried out by using arbitrary polygonal meshes. When a crack propagates, the remeshing operation involves only a small region of polygons around the propagating crack tip. Mesh mapping technology is used to transfer the response parameters to newly generated nodes. The proposed method is validated by analysing the impact of two identical bars and the dynamic contact of crack faces in the Koyna Dam. The results show that the dynamic contact model can effectively simulate the opening and closing of crack faces, and the longer the crack is, the greater the influence of the non-linearity of the contact along a crack face on the dynamic stress intensity factor and dynamic response. Finally, a dynamic crack propagation problem is simulated. Compared with the dynamic crack response without considering the contact constraint, when considering the contact constraint, the crack initiation angle is closer to the corresponding theoretical solution, and the crack propagation path is closer to the corresponding experimental result. Thus, the dynamic contact constraint condition of crack faces has an important influence on crack propagation.

Reducing conservatism in robust stability analysis of fractional-order-polytopic systems

This paper studies the robust stability of the fractional-order (FO) LTI systems with polytopic uncertainty. Generally, the characteristic polynomial of the system dynamic matrix is not an affine function of the uncertain parameters. Consequently, the robust stability of the uncertain system cannot be evaluated by well-known approaches including LMIs or exposed edges theorem. Here, an over-parameterization technique is developed to convert the main characteristic polynomial into a set of local over-parameterized characteristic polynomials (LOPCPs). It is proved that the robust stability of LOPCPs implies the robust stability of the uncertain system. Then, an algorithm is proposed to explore system’s robust stability through investigating the robust stability of these LOPCPs based on the exposed edges idea. For the sake of feasibility comparison, extensive examples are elaborated that reveal the superiority of the proposed algorithm.

Resilient Sensor Placement for Kalman Filtering in Networked Systems: Complexity and Algorithms

Given a linear dynamical system affected by noise, we study the problem of optimally placing sensors (at design-time) subject to a sensor placement budget constraint in order to minimize the trace of the steady-state error covariance of the corresponding Kalman filter. While this problem is NP-hard in general, we consider the underlying graph associated with the system dynamics matrix, and focus on the case when there is a single input at one of the nodes in the graph. We provide an optimal strategy (computed in polynomial-time) to place the sensors over the network. Next, we consider the problem of attacking (i.e., removing) the placed sensors under a sensor attack budget constraint in order to maximize the trace of the steady-state error covariance of the resulting Kalman filter. Using the insights obtained for the sensor placement problem, we provide an optimal strategy (computed in polynomial-time) to attack the placed sensors. Finally, we consider the scenario where a system designer places the sensors under a sensor placement budget constraint, and an adversary then attacks the placed sensors subject to a sensor attack budget constraint. The resilient sensor placement problem is to find a sensor placement strategy to minimize the trace of the steady-state error covariance of the Kalman filter corresponding to the sensors that survive the attack. We show that this problem is NP-hard, and provide a pseudo-polynomial-time algorithm to solve it.

Model Updating Using Frequency Response Functions Based on Sherman–Morrison Formula

Model updating plays an important role in dynamics modeling with high accuracy, which is widely used in mechanical engineering. In this paper, a model updating method using frequency response function (FRF) is proposed based on Sherman–Morrison formula, in which only the initial FRFs and parameter perturbations are employed to calculate the sensitivity avoiding repeated finite element (FE) analyses and improving the computational efficiency. Firstly, the sensitivity of FRFs to the design parameters is calculated by Sherman–Morrison formula based on the QR decomposition of the system dynamic stiffness matrix variation after parameter perturbations, then the influence of damping on the amplitude of FRFs is considered to select an appropriate frequency range, and finally conduct the model updating according to the sensitivity of the FRFs. By employing simulation examples of a truss and a solar wing and the experiment of an aluminum frame, the updating error is still within ±1.00% in the condition of 5% random white noise, which shows the proposed method has high accuracy and a certain anti-noise capability. When only a few numbers of frequency points are selected near the resonance peak of the FRFs, the result shows that selecting the appropriate frequency range and points can reduce the computational cost. The results of the experiment study show that the proposed method can effectively identify the structural parameters. The above results verify the feasibility and effectiveness of proposed model updating method using FRFs.

Application of Sub-Predictor Control for Linear Delayed Stochastic Systems

The problem of robust predictor-based H∞ state-feedback control for uncertain continuous-time stochastic state-multiplicative retarded systems is extended to include a set of sub-predictors, thus considerably improving the control performance. The multiplicative noise appears in the system dynamic matrix and in the measurement matrix of the observed signal, while the delay resided in the input to the system. In this problem, a cost function is defined which is the expected value of the standard H∞ performance index with respect to the uncertain parameters. In the robust case, the solution is obtained via a simple set of linear matrix inequalities. We bring a detailed numerical example that demonstrates the superiority of designing state-feedback control via a set of sub predictor compared to a single predictor design.

Kinematics and workspace analysis of a spherical exoskeleton parallel mechanism

Based on the biomechanical simulation curve of OpenSim, an open source software of biomechanical model, a spherical exoskeleton parallel mechanism with two degrees of freedom for hip joint is proposed in this paper for the rehabilitation therapy of patients with impaired leg motor function or elderly people with walking dysfunction. Firstly, the parallel mechanism is modeled and the position inverse solution of the parallel mechanism is obtained using inverse kinematics analysis. The velocity analysis expression of the mechanism is derived by deriving the inverse kinematics solution. The model is imported into the mechanical system dynamics analysis software ADAMS and matrix processing analysis software MATLAB to carry out simulation experiments. The correctness of the velocity analysis is verified by comparing the velocity simulation results of the two methods. Then, three singular types of the mechanism are analyzed according to the obtained Jacobian matrix. According to the inverse solution of the mechanism, the reachable workspace of the mechanism is obtained by programming in MATLAB with given mechanism parameters and restriction conditions. Finally, the prototype platform is built. The experimental results show that the exoskeleton hip joint using this parallel mechanism can satisfy the requirement of rotation angle of human hip joint movement, but also can be good to assist patients with leg flexion-extension movement and adduction-abduction movement, and it is helpful to carry out corresponding rehabilitation training. It also has theoretical significance and application value for the research work of human hip exoskeleton parallel mechanism.

Об исследовании малых колебаний сложных электрических систем

The mathematical model of a controlled multimachine electric power system in matrix form is developed. The model is based on space state equations and on using the system nesting technology. By using the obtained mathematical model, it is possible to investigate small-signal stability of a controlled complex electric system through determining the eigenvalues of the system dynamics matrix. The method of decomposing the complex electric system initial model by means of semi-orthogonal matrix zero dividers is applied for shifting the system poles to the desired positions.

Optimal jamming attack schedule for remote state estimation with two sensors

In cyber-physical systems (CPS), cyber threats emerge in many ways which can cause significant destruction to the system operation. In wireless CPS, adversaries can block the communications of useful information by channel jamming, incurring the so-called denial of service (DoS) attacks. In this paper, we investigate the problem of optimal jamming attack scheduling against remote state estimation wireless network. Specifically, we consider that two wireless sensors report data to a remote estimator through two wireless communication channels lying in two unoverlapping frequency bands, respectively. Meanwhile, an adversary can select one and only one channel at a time to execute jamming attack. We prove that the optimal attack schedule is continuously launching attack on one channel determined based on the system dynamics matrix. The theoretical results are validated by numerical simulations.

Learning Predictive State Representations From Non-Uniform Sampling

Predictive state representations (PSR) have emerged as a powerful method for modelling partially observable environments. PSR learning algorithms can build models for predicting all observable variables, or predicting only some of them conditioned on others (e.g., actions or exogenous variables). In the latter case, which we call conditional modelling, the accuracy of different estimates of the conditional probabilities for a fixed dataset can vary significantly, due to the limited sampling of certain conditions. This can have negative consequences on the PSR parameter estimation process, which are not taken into account by the current state-of-the-art PSR spectral learning algorithms. In this paper, we examine closely conditional modelling within the PSR framework. We first establish a new positive but surprisingly non-trivial result: a conditional model can never be larger than the complete model. Then, we address the core shortcoming of existing PSR spectral learning methods for conditional models by incorporating an additional step in the process, which can be seen as a type of matrix denoising. We further refine this objective by adding penalty terms for violations of the system dynamics matrix structure, which improves the PSR predictive performance. Empirical evaluations on both synthetic and real datasets highlight the advantages of the proposed approach.

Optimal synthesis of the low-order model parameters for a large-scale energy system on the basis of the gramians spectral decomposition

Abstract The structure of the low-order model is characterized by the fact the eigenvalues of its dynamics matrix coincide with the weakly stable eigenvalues of the system, which are calculated using the spectrum of the system dynamics matrix and the method of the gramians spectral decompositions. The optimal synthesis of the simplified model parameters is based on the approximate solution of the matrix polynomial equation for the matrices of the numerator of the system matrix transfer function and its simplified model. The solution of the synthesis problem leads to the solution of the linear matrix algebraic equation formed by the adaptability matrices. The energy functionals of the system and its simplified model can be determined by means of spectral decompositions of the squared H2 -norm of the matrix transfer function of the system and its simplified model. When the system approaches to the stability boundary, the stability margins of the system and its simplified model determined by the dominant weakly stable eigenvalues coincide asymptotically. This makes it possible not only to guarantee the stability of the simplified model, but also to determine the stability margins of a high-order system with help of the stability margins of a low-order model. The method makes it possible to estimate the oscillational instability caused by the resonant interaction of intraregional and inter-area oscillations and also to evaluate the impact of the input and output system matrices on the system stability using input and output matrices of its simplified model.

A Distributed Framework for Stability Evaluation and Enhancement of Inverter-Based Microgrids

In this paper, a distributed framework is established to evaluate and improve the small-disturbance stability of a microgrid with inverter-based distributed generators (DGs). First, we propose a distributed protocol such that a selected pilot DG bus collects necessary data from finite steps of communication between physically neighboring buses. Then, we design an algorithm that enables the pilot DG bus to reconstruct the system dynamic Jacobian matrix from the data and carry out stability evaluation locally. Further, if the evaluation result does not meet a preset requirement, we develop a distributed successive linear programming algorithm to enhance the system stability via sparse communication among a minority of DG buses. The proposed algorithms have high efficiency and versatility—they apply to lossy microgrids with either radial or meshed topology under various operation scenarios. Moreover, the control actions for stability enhancement preserve the predefined economic dispatch and power sharing schemes. The results are validated by the case study of a 14-bus microgrid.

Method for operating mode identification of multi‐function radars based on predictive state representations

Operating mode identification of multi-function radars (MFRs) is critical in radar threat evaluation and electronic jamming decisions. The hidden Markov model (HMM) in conventional methods relies on the MFR prior knowledge and cannot accurately describe the MFR signal sequence with regularity. An identifying method is presented that is based on predictive state representation (PSR) models without the help of prior information, which is unlikely to be obtained. Simulations of synthetic MFR signals are used to demonstrate the applicability of the novel method and to summarise the training parameter setting principles, which are of practical significance in applications of the method. The PSR-based method is shown to be more effective than the HMM-based methods, in particular when applied to the radar signal sequence with unknown regularity. The method also contains a novel approach to reducing the dimension of the system-dynamics matrix, which enables the algorithm to perform well in conditions of heavy noise. Simulation results attest to the validity of the proposed method. With the proposed method, the operating modes of radars can be identified accurately, which is significant in supporting adaptive countermeasures against MFRs.

Move Suppression Calculations for Well-Conditioned MPC

Several popular tuning strategies applicable to Model Predictive Control (MPC) schemes such as GPC and DMC have previously been developed. Many of these tuning strategies require an approximate model of the controlled process to be obtained, typically of the First Order Plus Dead Time type. One popular method uses such a model to analytically calculate an approximate value of the move suppression coefficient to achieve a desired condition number for the regularized system dynamic matrix; however it is not always accurate and tends to under-estimate the required value. In this paper an off-line method is presented to exactly calculate the move suppression coefficient required to achieve a desired condition number directly from the unregularized system dynamic matrix. This method involves an Eigendecomposition of the system dynamic matrix - which may be too unwieldy in some cases –and a simpler analytical expression is also derived. This analytical expression provides a guaranteed tight upper bound on the required move suppression coefficient yielding a tuning formula which is easy to apply, even in on-line situations. Both methods do not require the use of approximate or reduced order process models for their application. Simulation examples and perturbation studies illustrate the effectiveness of the methods in both off-line and on-line MPC configurations. It is shown that accurate conditioning and improved closed loop robustness can be achieved.

Feasibility investigation of large-scale model suspended by cable-driven parallel robot in hypersonic wind tunnel test

Cable-driven parallel robot is a special kind of robot, which is actuated by cables. It is already applied in the low speed wind tunnel to get aerodynamic measurement of aircraft model, and the aircraft pose could be adjusted by changing the cable length. Whether it can be used in hypersonic wind tunnel still needs further discussion. This paper presents the dynamics and aerodynamics analysis of a large-scale model supported by 6-DOF cable-driven parallel robot to investigate the feasibility of this special kind of suspension system in hypersonic wind tunnel. The description of this setup with a X-51A-like model is given, and then based on the system dynamic equations, aerodynamic force and stiffness matrix are derived. In the simulation, properties of dynamics and aerodynamics are mainly concerned. A typical shock tunnel with flow duration of about 100 milliseconds is taken as an example, and results show that the system is stable enough to meet the fundamental static wind tunnel test. From the cable tension variation under impact load and the sensitivity analysis, it is likely accessible to derive the aerodynamic forces. Compared with the sting suspension method, cable-driven parallel robot has the priority of higher inherent frequency and more flexible degrees. The interference to the flow field induced by cables is also preliminarily proved to be small by the CFD simulation, which can be acceptable and corrected. Researches conducted show the feasibility of cable-driven parallel robot’s application in hypersonic wind tunnel.

Prediction of L70 Life and Assessment of Color Shift for Solid-State Lighting Using Kalman Filter and Extended Kalman Filter-Based Models

Solid-state lighting (SSL) luminaires containing light-emitting diodes (LEDs) have the potential of seeing excessive temperatures when being transported across country or being stored in nonclimate controlled warehouses. They are also being used in outdoor applications in desert environments that see little or no humidity but will experience extremely high temperatures during the day. This makes it important to increase our understanding of what effects high temperature exposure for a prolonged period of time will have on the usability and survivability of these devices. Traditional light sources “burn out” at the end of life. For an incandescent bulb, the lamp life is defined by B50 life. However, the LEDs have no filament to “burn“. The LEDs continually degrade, and the light output decreases eventually below useful levels causing failure. Currently, the TM-21 test standard is used to predict the L70 life of LEDs from LM-80 test data. Several failure mechanisms may be active in a LED at a single time, causing lumen depreciation. The underlying TM-21 model may not capture the failure physics in presence of multiple failure mechanisms. The correlation of lumen maintenance with underlying physics of degradation at system level is needed. In this paper, Kalman filter (KF) and extended Kalman Filters (EKF) have been used to develop a 70% Lumen Maintenance Life Prediction Model for LEDs used in SSL luminaires. Ten-thousand-hour LM-80 test data for various LEDs have been used for model development. System state at each future time has been computed based on the state space at preceding time step, system dynamics matrix, control vector, control matrix, measurement matrix, measured vector, process noise, and measurement noise. The future state of the lumen depreciation has been estimated based on a second-order Kalman Filter model and a Bayesian Framework. The measured state variable has been related to the underlying damage using physics-based models. Life prediction of L70 life for the LEDs used in SSL luminaires from KF- and EKF-based models have been compared with the TM-21 model predictions and experimental data.

An H∞ LPV control for a class of LPV systems using a descriptor approach: Application to a wind turbine mode

This paper is devoted to the synthesis of an H∞ LPV state feedback controller, applied to a wind turbine which is described by an LPV model. Firstly, we focus on the modeling of wind turbines, to propose a model that describes as good as possible the system behaviour. A new approach that allows us, via LMIs, to design an LPV controller for a class of LPV systems is then presented. The interest of the approach is to decouple the system dynamic matrix and the lyapunov one using a desriptor approach. Finally, to illustrate the proposed control strategy, some simulation results are presented.

Dynamic sensor transmission power scheduling for remote state estimation

In this paper, we consider the problem of sensor transmission power scheduling for remote state estimation. We assume that the sensor has two transmission energy levels, where the high level corresponds to a high packet reception ratio. By exploiting the feedback information from the remote estimator, we aim to find an optimal transmission power schedule. We formulate the problem as a Markov decision process, and analytically develop a simple and optimal dynamic schedule which minimizes the average estimation error under the energy constraint. Furthermore, we derive the necessary and sufficient condition under which the remote state estimator is stable. It is shown that the estimation stability only depends on the high-energy packet reception ratio and the spectral radius of the system dynamic matrix.