Second-order Linear Systems Research Articles

Insights into instability of friction-induced vibration of multi-degree-of-freedom models

This study addresses a fundamental problem of linear friction-induced vibration (FiV), the intriguing role of damping on the instability of FiV in the form of mode-coupling. Two well-known classic 2-degree-of-freedom (DoF) slider-belt models are revisited, and a 4-DoF slider-belt model considering different damping configurations in the slider and the belt is investigated. Since proportional damping is very rare in reality, non-proportional damping is given much attention in this paper.Firstly, the analytical solutions of bifurcation boundary of the critical coefficients of friction of the classic 2-DoF models are derived, which show the subtle role of damping clearly and theoretically explain and clear misunderstandings and confusions on the phenomenon about the drop of the stability boundary from that of the undamped system caused by adding non-proportional damping. Secondly, a further increase of non-proportional damping is found to improve the stability and even bring higher degree of stability than increasing proportional damping (which is a new finding). Thirdly, the influences of the different damping distributions among the system components on the stability of the 4-DoF slider-belt model are explored. The evolution of the bifurcation boundary indicates the importance of the ratio of the horizontal to the vertical damping in the slider and in the belt. For some non-proportional damping values, the bifurcation boundary fluctuates, which could exhibit local optimal values of the ratio of damping in different directions of a system component and can be exploited in design against FiV. These findings are valid for general linear second-order dynamic systems with an asymmetric stiffness matrix.

Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions

<p style='text-indent:20px;'>The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.</p>

SVD-Krylov based techniques for structure-preserving reduced order modelling of second-order systems

<abstract><p>We introduce an efficient structure-preserving model-order reduction technique for the large-scale second-order linear dynamical systems by imposing two-sided projection matrices. The projectors are formed based on the features of the singular value decomposition (SVD) and Krylov-based model-order reduction methods. The left projector is constructed by utilizing the concept of the observability Gramian of the systems and the right one is made by following the notion of the interpolation-based technique iterative rational Krylov algorithm (IRKA). It is well-known that the proficient model-order reduction technique IRKA cannot ensure system stability, and the Gramian based methods are computationally expensive. Another issue is preserving the second-order structure in the reduced-order model. The structure-preserving model-order reduction provides a more exact approximation to the original model with maintaining some significant physical properties. In terms of these perspectives, the proposed method can perform better by preserving the second-order structure and stability of the system with minimized $ \mathcal{H}_2 $-norm. Several model examples are presented that illustrated the capability and accuracy of the introducing technique.</p></abstract>

Transitivity of Commutativity for Linear Time-Varying Physical Systems

In this contribution, the transitivity property of commutative first-order linear time-varying systems with and without initial conditions is investigated. It is proven that transitivity property of first-order systems holds with and without initial conditions. On the base of impulse response function, transitivity of commutation property is formulated for any triplet of commutative linear time-varying relaxed systems. Transitivity proves are given for some special combinations of first and second-order linear time-varying systems which are initially relaxed.

A Note on the Usefulness of Constrained Fourth-Order Latent Differential Equation Models in the Case of Small T

Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., T=50), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a data set, which consists of T=56 observations of daily stress for N=44 individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T. This is of great importance, as parameter estimates depend on the embedding dimension as well as on the latent differential equations model. Consequently, the wavelength as quantity of potential substantive interest may vary considerably. We extend the modeling approaches used in past research by including multiple subjects, by accounting for individual differences in equilibrium, and by including multiple instead of one single observed indicator.

Symbolic finite element discretization and model order reduction of a multiphysics model for IPMC sensors

The multiphysics model of ionic polymer-metal composite (IPMC) sensors proposed by Zhu has a significant advantage of being able to describe the dynamic sensor response, which highly depends on humidity, by explicitly considering solvent dynamics. However, it is difficult to perform analysis and simulation because Zhu’s model is represented by complex non-linear partial differential equations. This paper describes the symbolic finite element discretization of Zhu’s model and further discusses the essential dynamics of the reduced-order model extracted from the finite element model. The obtained linear ordinary differential equations, or the state equation, can be easily implemented in simulators via common programming languages. The simulation results of an in-house simulator implemented by MATLAB code show good agreement with those of direct numerical simulation by using commercial software, COMSOL. To further simplify the model, the minimum order required for an appropriate approximation is numerically investigated by using a model order reduction technique. This paper reveals that the dynamic response of an IPMC sensor can be consequently approximated by a first-order or second-order linear time-invariant system.

Construction of Lyapunov Functions for Second-Order Linear Stochastic Stationary Systems

Construction of Lyapunov Functions for Second-Order Linear Stochastic Stationary Systems

Uniformly asymptotic stability of second-order linear time-varying systems

This paper is concerned with the stability of second-order linear time-varying systems. By utilizing the Lyapunov approach, a generally uniformly asymptotic stability criterion is obtained by adding the system matrices into the quadratic Lyapunov candidate function. In the case of the derivative of the Lyapunov candidate function is semi-positive definite, the stability criterion is also efficient. Based on the proposed results, the systems with uncertain disturbances such as structured independent and structured dependent perturbations are considered. Using the matrix measure and the singular value theory, the bounds of the uncertainties are obtained that guarantee the system uniformly asymptotically stable, while the bounds of state feedback control input are also derived to stabilize the second-order linear time-varying systems. Finally, several numerical examples are given to prove the validity and correctness of the proposed criteria with existing ones.

Receptance-based robust stability criteria for second-order linear systems with time-varying delay and unstructured uncertainties

This paper presents new frequency domain criteria for the robust stability analysis of closed-loop second-order systems under time-varying internal delay. The proposed approach is derived from the small gain theorem for systems described by the versatile system receptance to guarantee robust stability despite time-varying delays and model uncertainties. Robust stability of systems with time-varying delays is challenging to cope because time-invariant paradigm does not hold, but the proposed approach is directly based on the frequency response analysis of the closed-loop system with unstructured uncertainty bound. The criteria can be applied to either single or multiple inputs systems, and no constraint is imposed on the delay derivative. A simple receptance-based Internal Model Control strategy is proposed to improve robust stability margins while nominal state-feedback design is preserved. Thus, a controller obtained from the delay-free nominal model can be used to ensure the robust stability of the uncertain second-order system with time-varying delay. Simulation case studies are used to illustrate the benefits of the proposed criteria.

Polynomial Controller Design and Its Application: Experimental Validation on a Laboratory Setup of Nonideal DC–DC Buck Converter

The transient response control is one of the most important tasks in the control system design. The transient response has characterized based on the relationship between characteristic ratios and time domain specifications. The characteristic ratios are the design parameters of the characteristic ratio assignment (CRA) approach. This article presents a novel polynomial controller for all types of linear second-order systems. The proposed controller is designed using CRA. The important features of this controller are that it has simple design steps, set-point tracking, and disturbance rejection capabilities. Initially, the proposed controller is designed for an unstable nonminimum-phase second-order system to demonstrate the performance and robustness. Subsequently, the presented controller is analyzed and validated on the nonideal dc–dc buck converter. The buck converter is highly sensitive to the suddenly changing load conditions. The major issue of output voltage control in the dc–dc buck converter is addressed in this article. In addition, the robustness and performance of the proposed controller are verified for the sudden change in supply input voltage, the sudden change in the reference voltage, the parametric uncertainty in inductor ( $L$ ) and capacitor ( $C$ ), and the sudden change in load. Furthermore, the efficacy of the proposed controller is evaluated by comparing it with the recently published control schemes for this converter. The proposed controller is implemented using dSPACE 1104 real-time controller. Applicability of the presented controller is corroborated experimentally on the nonideal dc–dc buck converter. The simulation as well as hardware results recommend that the proposed controller is suited for the nonideal dc–dc buck converter under such conditions.

Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

Improved DC-Link Voltage Regulation Strategy for Grid-Connected Converters

In this article, an improved dc-link voltage regulation strategy is proposed for grid-connected converters applied in dc microgrids. For the inner loop of the grid-connected converter, a voltage modulated direct power control is employed to obtain two second-order linear time-invariant systems, which guarantees that the closed-loop system is globally exponentially stable. For the outer loop, a sliding mode control strategy with a load current sensor is employed to maintain a constant dc-link voltage even in the presence of constant power loads at the dc-side, which adversely affect the system stability. Furthermore, an observer for the dc-link current is designed to remove the dc current sensor at the same time improving the reliability and decreasing the cost. From both simulation and experimental results obtained from a 15-kVA prototype setup, the proposed method is demonstrated to improve the transient performance of the system and has robustness properties to handle parameter mismatches compared with the input-output linearization method.

Matrix Laplace Transform Method and It Applications on Spring-Mass Systems

There are several methods to solve an initial value problem of second-order homogenous linear systems of differential equations with constant coefficients. That are elimination method and matrix method. Whereas to solve nonhomogenous systems, used undetermined coefficient method and variation of parameter method, that through some difficulties and complex procceses. But then, there is an alternative method to solve it. It is Matrix Laplace Transform Method. The goals of the research are to explain Matrix Laplace Transform Method and use it to solve initial value problems of second-order homogenous linear systems of differential equations with constant coefficients. All of matrix entries are constant. The result of the research is be obtained solutions of second-order linear systems of differential equations with constant coefficients use Matrix Laplace Transform Method

Minimum attention stochastic control with homotopy optimization

A method of designing control inputs for tracking problems when models considered in the form of stochastic differential equations is proposed. Using the ideas of control vector parameterization, the control input identification problem is formulated as a parameter identification problem, which is solved using homotopy optimization. To further obtain a control with the least number of switchings, i.e., minimum attention control, a sparse recovery framework is developed. The accuracy of the proposed methods is demonstrated with the help of a linear second-order system and a non-linear quadruple tank system.

On Containment for Linear Systems With Switching Topologies: A Novel State Transition Matrix Perspective.

This article studies the containment control problem for a group of linear systems, consisting of more than one leader, over switching topologies. The input matrices of these linear systems are not required to have full-row rank and the switching can be arbitrary, making the problem quite general and challenging. We propose a novel analysis framework from the viewpoint of a state transition matrix. Specifically, according to the inherent linearity, we successfully establish a connection between state transition matrices of the above multileader system and a virtual leader-following system obtained by combining those leaders. This enlightening result relates the containment problem to a consensus one. Then, by analyzing the property of the state transition matrix, we uncover that each component of any follower's state converges to the convex hull spanned by the corresponding components of the leaders', provided some mild conditions are satisfied. These conditions are derived in terms of the concept of a positive linear system. A special case of the second-order linear system is further discussed to illustrate these conditions. Moreover, two different design methods of the feedback gain matrix are provided, which additionally require that the network topology contains a united spanning tree all the time.

Parametric uncertain second-order linear system output-adaptive stabilization: An integral and MRCA based approach

This work proposes a solution for the adaptive stabilization of second-order linear systems with unknown parameters using only position measurements. Only the sign of the control law is known and the system is affected by constant perturbations. The crux of the proposed control scheme the use of an alternative representation of the system obtained through integrating over time the original system representation. In this way, the whole state of the alternative representation is available from measurements, and an extended state space representation is obtained through the alternative representation and the differential equation related to the integral over time of the original input. Later, an Adaptive Control methodology and an backstepping procedure are used to establish closed-loop stability. Simulation results support the findings.

TheFinite Element Modeling and Experimental Study of Sandwich Plates with Frequency-Dependent Viscoelastic Material Model.

Athree-layer composite plate element is developed for finite element modeling and vibration analysis of sandwich plate with frequency-dependent viscoelastic material core. The plate element is quadrilateral element bounded by four-node with 7-degree-of-freedom per node. The frequency-dependent characteristics of viscoelastic material parameters are described using the Biot model. The method of identifying the parameters of the Biot model is given. By introducing auxiliary coordinates, the Biot model is combined with the finite element equation of the viscoelastic sandwich plate. Through a series of mathematical transformations, the equation is transformed into a standard second-order steady linear system equation form to simplify the solution process. Finally, the vibration characteristics of the viscoelastic sandwich plate are analyzed and experimentally studied. The results show that the method in this paper is correct and reliable, and it has certain reference and application value for solving similar engineering vibration problems.

Star-shaped control-vector sets of second-order systems with PWM-type input

This paper deals with second-order continuous-time linear time-invariant systems by means of a pulse-width modulation (PWM)-type input. In general, it is difficult to analyze and design a PWM system because of its nonlinearity in terms of the duty ratio. In this paper, it is shown that the nonlinearity appearing in the discrete-time model at the sampling instants can be eliminated by varying the location of the pulse as well as the duty ratio.

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

AC Tie-Line Power Oscillation Mechanism and Peak Value Calculation for a Two-Area AC/DC Parallel Interconnected Power System Caused by LCC-HVDC Commutation Failures

With the rapid development of ultra-high-voltage (UHV) AC/DC, especially the step-by-step upgrading of the UHV DC transmission scale, security presents new challenges. Commutation failure (CF) is a common fault in line commutated converter (LCC) high-voltage direct current (HVDC) power systems. Once failure happens, it may cause power oscillations in a system. In this paper, taking a two-area AC/DC parallel interconnected power system as the example, based on the impulse response model of second-order linear system, the mechanism of power oscillation on the AC tie-line caused by CF are clarified. It is proved that the peak value of the AC tie-line power oscillation is mainly determined by the DC power and the equivalent CF duration, the frequency and damping ratio of dominant area oscillation mode. Meanwhile, the peak time is mainly determined by the oscillation frequency. Finally, the correctness and effectiveness of the algorithms are verified by a simulation analysis of an extended IEEE-39-bus AC/DC parallel interconnected power system. These research results can provide a basis for the arrangement of the operating modes and the formulation of control measures for interconnected power grids.