Order Linear Filter Research Articles

Stochastic Roll Dynamics of Smooth and Impacting Vessels in Random Waves

Ship instability is frequently associated with extreme roll motion. In this paper smooth and vibro-impact dynamics of vessels rolling under random wave excitations including nonlinearities is investigated. The vessel is represented by a single degree of freedom roll model. The random excitation due to waves and ice is modeled by a sum of random pulses with Poisson arrival rate. Stochastic P-bifurcation analysis is carried out by examining the variation with parameter of the joint probability density functions (jpdf) of the response computed numerically by the solution of the corresponding Fokker–Planck–Kolmogorov (FPK) equation. The finite element (FE), finite difference (FD), path integral (PI) and radial basis neural (RBN) network formulation methods are used in the solution of the FPK equation. The D-bifurcation analysis is analyzed by computing the largest Lyapunov exponent. The mean upcrossing rate is computed using Rice’s formula. The random wave excitation is also modeled as the response of a second order linear filter to white noise and the random response of the ship is investigated by solution of the corresponding FPK equation of the four dimensional Markov system by the FD method. Non-smooth coordinate transformations like the Zhuravlev and Ivanov transformations are used converting the discontinuous systems to equivalent smooth systems in the impact dynamics analysis. The adaptive time stepping method (ATSM) approach is used to accurately locate the discontinuity point and a Brownian tree approach is used to direct the integration along the correct Brownian path. Different models for nonlinear damping and the restoring moment are considered and their effects on ship stochastic response such as chaotic motion, unbounded rotational motion, impact modulation motion and ship capsizing are investigated.

Control of single input Hamiltonian systems based on the flatness of their tangent linearization

We explore the incremental flatness based control of single input under-actuated nonlinear Hamiltonian systems exhibiting a controllable tangent linearization model around a given equilibrium point. General properties of controllable linearized Hamiltonian systems are presented, which significantly ease the stabilizing, or output reference trajectory tracking, feedback controller design for the nonlinear system. Controllability of the tangent linear system is equivalent to its flatness. A flat filter controller and a set of nested controllers are presented which are based on incremental position measurements alone. There will be no need for explicit estimations of the conjugate momenta through observers. The proposed controller is constituted by a set of nested, second order linear Flat Filter compensation networks acting on the second order pure integration models, naturally present in the input-to-flat output system structure. For pure integration perturbed systems, Flat Filters have been shown to be equivalent, in general, to reduced order extended state observer based Active Disturbance Rejection controllers. A challenging stabilization problem for an unstable, nonlinear, two spring–mass system carrying an inverted pendulum is considered. Computer simulations illustrate the effectiveness of the proposed controller design in the presence of flat output measurement noise.

Lyapunov Function based Criteria for Ship Rolling in Random Beam Seas

Abstract The aim of this study is to present a Lyapunov function which can be used to derive an intact stability criterion for a ship in random beam seas. First, the mathematical model of the rolling motion of ships in random beam seas is introduced. The random wave excitation is described by a spectrum which is depended on the wave energy spectrum and the amplitude of the moment of roll. This spectrum is generated by a second order linear filter. Second, the methodology of creating a Lyapunov function is explained briefly. Then, there is outlined the way by which Lyapunov function can be used as the intact stability criterion for a ship. The proposed criterion is derived by considering the weather criteria for German naval vessels. Finally, the coherence of the boundary of safe basin obtained by Lyapunov function with the numerical results obtained by Euler-Maruyama Method is presented. From the results it can be deduced that the Lyapunov function can be used to define an intact stability criterion.

Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows

Recent progress in the improvement of numerical stability and accuracy of the Yee and Sjögreen [49] high order nonlinear filter schemes is described. The Yee & Sjögreen adaptive nonlinear filter method consists of a high order non-dissipative spatial base scheme and a nonlinear filter step. The nonlinear filter step consists of a flow sensor and the dissipative portion of a high resolution nonlinear high order shock-capturing method to guide the application of the shock-capturing dissipation where needed. The nonlinear filter idea was first initiated by Yee et al. [54] using an artificial compression method (ACM) of Harten [12] as the flow sensor. The nonlinear filter step was developed to replace high order linear filters so that the same scheme can be used for long time integration of direct numerical simulations (DNS) and large eddy simulations (LES) for both shock-free turbulence and turbulence-shock waves interactions. The improvement includes four major new developments: (a) Smart flow sensors were developed to replace the global ACM flow sensor [21,22,50]. The smart flow sensor provides the locations and the estimated strength of the necessary numerical dissipation needed at these locations and leaves the rest of the flow field free of shock-capturing dissipation. (b) Skew-symmetric splittings were developed for compressible gas dynamics and magnetohydrodynamics (MHD) equations [35,36] to improve numerical stability for long time integration. (c) High order entropy stable numerical fluxes were developed as the spatial base schemes for both the compressible gas dynamics and MHD [37,38]. (d) Several dispersion relation-preserving (DRP) central spatial schemes were included as spatial base schemes in the framework of our nonlinear filter method approach [40]. With these new scheme constructions the nonlinear filter schemes are applicable to a wider class of accurate and stable DNS and LES applications, including forced turbulence simulations where the time evolution of flows might start with low speed shock-free turbulence and develop into supersonic speeds with shocks. Representative test cases for both smooth flows and problems containing discontinuities for compressible flows are included.

Efficient Monte Carlo simulation and Grim effective wave model for predicting the extreme response of a vessel rolling in random head seas

In this paper, a computationally efficient Monte Carlo simulation (MCS) approach is introduced in order to estimate the extreme response statistics of parametric rolling in random longitudinal seas. Basically, parametric roll is caused by sufficiently large oscillations of the roll restoring moment occurring within certain frequencies of wave encounter (approximately twice the natural roll frequency). The concept of the Grim effective wave is applied herein in order to approximate the variation of the restoring moment in random waves. A fourth order linear filter is introduced to model the random effective wave amplitude process, which is assumed to be the driving process for the variation of the restoring term as well as for the stochastic nonlinear system of the roll motion by application of the Grim effective wave approximation. For the stochastic dynamical system, the roll response is a random process and an extrapolation procedure is developed for estimating the extreme values of the response statistics by assuming regular behavior in the tail region of the mean upcrossing rate. The rationality of the linear filter model and the feasibility of an efficient MCS method based on extrapolation techniques for predicting the extreme roll response are illustrated. Furthermore, the phenomenon of parametric roll in random seas as well as the effect of vessel speed on the stochastic roll response are investigated.

Stochastic roll response for a vessel with nonlinear damping models and steady heeling angles in random beam seas

Loss of ship stability is most frequently associated with extreme roll motion. For the case of random beam sea, a single-degree-of-freedom (SDOF) model is applied in the present study in order to represent the stochastic rolling behavior. The random wave excitation term in the SDOF model is approximated as a filtered white noise process by applying a second order linear filter. Accordingly, the original SDOF model is extended into a four-dimensional (4D) dynamic system. The coupled dynamic system can be viewed as a Markov system whose probabilistic properties are governed by the corresponding Fokker–Planck equation. Based on the convenient Markov property, a host of useful response statistics can be obtained by an efficient path integration (PI) method. Different nonlinear damping models, i.e. the linear-plus-quadratic damping (LPQD) model and the linear-plus-cubic damping (LPCD) model, and their effects on the stochastic roll response are investigated and the influence of the steady heeling angle on the response level associated with ship rolling in random seas is also studied. Furthermore, the accuracy of the response statistics computed by the PI technique is verified by means of the versatile Monte Carlo simulation (MCS) technique.

Numerical dissipation control in high order shock-capturing schemes for LES of low speed flows

The Yee & Sjögreen adaptive numerical dissipation control in high order scheme (High Order Filter Methods for Wide Range of Compressible Flow Speeds, ICOSAHOM 09, 2009) is further improved for DNS and LES of shock-free turbulence and low speed turbulence with shocklets. There are vastly different requirements in the minimization of numerical dissipation for accurate turbulence simulations of different compressible flow types and flow speeds. Traditionally, the method of choice for shock-free turbulence and low speed turbulence are by spectral, high order central or high order compact schemes with high order linear filters. With a proper control of a local flow sensor, appropriate amount of numerical dissipation in high order shock-capturing schemes can have spectral-like accuracy for compressible low speed turbulent flows. The development of the method includes an adaptive flow sensor with automatic selection on the amount of numerical dissipation needed at each flow location for more accurate DNS and LES simulations with less tuning of parameters for flows with a wide range of flow speed regime during the time-accurate evolution, e.g., time varying random forcing. An automatic selection of the different flow sensors catered to the different flow types is constructed. A Mach curve and high-frequency oscillation indicators are used to reduce the tuning of parameters in controlling the amount of shock-capturing numerical dissipation to be employed for shock-free turbulence, low speed turbulence and turbulence with strong shocks. In Kotov et al. (High Order Numerical Methods for LES of Turbulent Flows with Shocks, ICCFD8, Chengdu, Sichuan, China, July 14–18, 2014) the LES of a turbulent flow with a strong shock by the Yee & Sjögreen scheme indicated a good agreement with the filtered DNS data. A work in progress for the application of the adaptive flow sensor for compressible turbulence with time-varying random forcing is forthcoming. The present study examines the versatility of the Yee & Sjögreen scheme for DNS and LES of traditional low speed flows without forcing. Special attention is focused on the accuracy performance of this scheme using the Smagorinsky and the Germano–Lilly SGS models.

Finite Control Set Model Predictive Speed Control of a DC Motor

The paper describes the design procedure for a finite control set model predictive control (FCS-MPC) of brushed permanent magnet DC (PMDC) machine supplied from DC-DC converter. Full order linear Kalman filter is used for estimation of an unmeasured load torque and reduction of speed measurement noise. A new cost function has been introduced with a feedforward dynamic current component and a feedforward static load current component. The performance of the proposed control strategy is compared to the conventional PI-PWM cascade speed control through the experimental verification on the 250 W laboratory prototype. Obtained results show excellent dynamic behaviour and indicate possible energy savings of the proposed speed control.

Stochastic nonlinear ship rolling in random beam seas by the path integration method

Nonlinear ship rolling in random seas is a serious threat to ship stability. In this work, the dynamic stability of the vessel in random seas is evaluated by means of probabilistic methods. Specifically, the Markov diffusion theory is applied in order to study the stochastic aspects of the roll motion driven by random wave loads. The roll motion is modeled as a single-degree-of-freedom (SDOF) model in which the nonlinear damping and restoring terms as well as the random wave excitation are all incorporated. The stationary wave excitation moment in the SDOF model is represented as a filtered white noise by employing a second order linear filter. Therefore, a four-dimensional (4D) Markov dynamic system is established by combing the SDOF model with the linear filter model. Because the probabilistic property of the 4D Markov system is governed by the Fokker–Planck (FP) equation, the response statistics of roll motion can be obtained by solving the FP equation via an efficient 4D path integration (PI) method, which is based on the Markov property of the coupled dynamic system. Furthermore, the random wave excitation is approximated by an equivalent Gaussian white noise, and a two-dimensional (2D) PI technique is applied in order to obtain the response statistics of the dynamic system driven by this Gaussian white noise. The rationality and accuracy of applying the equivalent Gaussian white noise to simulate nonlinear ship rolling in random seas is studied. Moreover, the accuracy of the response statistics computed by the 4D and 2D PI techniques is verified by means of the versatile Monte Carlo simulation technique.

COVARIANCE ASSISTED SCREENING AND ESTIMATION.

Consider a linear model Y = X β + z, where X = Xn,p and z ~ N(0, In ). The vector β is unknown and it is of interest to separate its nonzero coordinates from the zero ones (i.e., variable selection). Motivated by examples in long-memory time series (Fan and Yao, 2003) and the change-point problem (Bhattacharya, 1994), we are primarily interested in the case where the Gram matrix G = X'X is non-sparse but sparsifiable by a finite order linear filter. We focus on the regime where signals are both rare and weak so that successful variable selection is very challenging but is still possible. We approach this problem by a new procedure called the Covariance Assisted Screening and Estimation (CASE). CASE first uses a linear filtering to reduce the original setting to a new regression model where the corresponding Gram (covariance) matrix is sparse. The new covariance matrix induces a sparse graph, which guides us to conduct multivariate screening without visiting all the submodels. By interacting with the signal sparsity, the graph enables us to decompose the original problem into many separated small-size subproblems (if only we know where they are!). Linear filtering also induces a so-called problem of information leakage, which can be overcome by the newly introduced patching technique. Together, these give rise to CASE, which is a two-stage Screen and Clean (Fan and Song, 2010; Wasserman and Roeder, 2009) procedure, where we first identify candidates of these submodels by patching and screening, and then re-examine each candidate to remove false positives. For any procedure β̂ for variable selection, we measure the performance by the minimax Hamming distance between the sign vectors of β̂ and β. We show that in a broad class of situations where the Gram matrix is non-sparse but sparsifiable, CASE achieves the optimal rate of convergence. The results are successfully applied to long-memory time series and the change-point model.

HTSC Cascaded Linear-Phase Filter with Quasi-Elliptic Response

In view of the difficulty of designing a high-order linear phase filter with prototype parameters directly, this paper proposes a new approach to rapidly construct a high order linear phase filter through cascading two same low-order linear phase filters. A twelve-order HTSC cascaded linear phase filter is fabricated on YBCO/LaAlO3/YBCO substrate with the dimension of 44.1mm ×31.4mm. The measured result shows that the filter has a 56 MHz passband at a center frequency of 1993 MHz. The minimum insertion loss in the passband is 0.59 dB. The group delay variation in passband is less than 5 ns over 78.5% of the filter bandwidth which is in good agree with the simulation result.

A delay decomposition approach to [formula omitted] filter design for stochastic systems with time-varying delay

This paper investigates the problem of L 2 – L ∞ filter design for a class of stochastic systems with time-varying delay. The addressed problem is the design of a full order linear filter such that the error system is asymptotically mean-square stable and a prescribed L 2 – L ∞ performance is satisfied. In order to develop a less conservative filter design, a new Lyapunov-Krasovskii functional (LKF) is constructed by decomposing the delay interval into multiple equidistant subintervals, and a new integral inequality is established in the stochastic setting. Then, based on the LKF and integral inequality, the delay-dependent conditions for the existence of L 2 – L ∞ filters are obtained in terms of linear matrix inequalities (LMIs). The resulting filters can ensure that the error system is asymptotically mean-square stable and the peak value of the estimation error is bounded by a prescribed level for all possible bounded energy disturbances. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Properties of time integration with first order filter damping

Introduction of algorithmic damping by increasing the parameter values in the Newmark algorithm leads to undesirable low-frequency damping and reduced order of accuracy. It is demonstrated, how these effects can be removed by introducing an extra damping term in the form of a first order linear filter. When the linear filter is discretized in time, the state variable associated with the filter can be eliminated, leading to a weighted average of the equations of motion at two consecutive times. The filter procedure contains the known versions of alpha weighted Newmark methods as special cases, but gives a different and improved weighting of the excitation terms. A complete analysis of the properties of the algorithm when used on linear systems is given, including the frequency response properties. It is demonstrated that the effect of ‘overshoot’ is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system, and analytic results are presented for the magnitude of the effect. Copyright © 2005 John Wiley & Sons, Ltd.

Stochastic Response of Offshore Structures via Statistical Cubicization

This paper deals with the stochastic response of single-degree-of-freedom structures with polynomial restoring force excited by random Morison’s forces with mean current. The problem is recast by expressing the excitation by means of a cubic polynomial of the wave elevation, which in turn is assumed as a stationary zero-mean Gaussian process, whose spectral density is given by the output of a cascade of two second order linear filters having a Gaussian white noise as primary excitation. Thus, Ito’s stochastic differential calculus becomes applicable, and the solution is pursued with a moment equation approach by using a suitable closure scheme. The results of the applications compare well with digital simulation.

Nonlinear Extensions of Classical Controllers Using Symbolic Computation Techniques: A Dynamical Systems Unifying Approach

AbstractIn previous works we reported the development of symbolic computation tools to automate the design of nonlinear state feedback controllers [1], nonlinear PID controllers [2], nonlinear lag‐lead compensators [3], and nonlinear obsevers [4] using the extended linearization method [5]. In this paper we show that a careful analysis of the state variables representation of these classical controllers indicates that all of them are particular cases of the nonlinear extension of a mth order linear filter, consisting of a kth order input derivative operator followed by an output mth order linear dynamical system. Using this two blocks decomposition approach, the design of nonlinear extensions of nth order controllers can be decomposed into two independent subalgorithms: a kth order PD controller algorithm, and a mth order state vector feedback algorithm. Hence, an appropriate, assembly of our symbolic computation tools NL Feedback and NLPID could, in principle, allow to use the extended linearization method to synthesize nonlinear extension of arbitrary nth order linear filter (controllers).

A vestibulo-ocular reflex with no head movement.

Eye movements were produced in an elasmobranch preparation by electrical stimulation of the horizontal canal ampullary nerves. A pseudorandom binary sequence of stimulus pulse trains was delivered bilaterally. Eye position during this stimulus was cross-correlated with the stimulus pattern to obtain a linear model of the response. Sums of exponential functions were fitted to the crosscorrelogram data to estimate time-constants and transfer functions. The data was examined in the frequency domain by using Fourier transformation. The response is accurately described by a second order linear filter, which is essentially a low pass filter with a cutoff at 0.22 Hz. This nearly two octaves below the cutoff frequency of the eye motor plant, which has been estimated by the same method. Our data shows that there is no central phase compensation or prediction which might offset the substantial delay in eye motor plant response. We hypothesise that the necessary phase compensation may be achieved by driving the vestibulo-ocular reflex with sensory neurons having a phase advance at high frequency.

Suspension Bounce Response of Canadian MAGLEV Vehicle Under Guideway Excitations—Part 2: Stochastic Modeling and Analysis

The investigation of Part I is extended to the case when the Canadian MAGLEV vehicle is subjected to a combination of periodic as well as dominant random disturbances from the guideways. Appropriate mathematical modeling of the input process is presented in terms of spectral density function which is further modeled as the output of a second order linear filter under a white noise input for easy analysis. The combined filter-vehicle stochastic equations are solved numerically and the suspension bounce acceleration behavior for different system parameters is presented.

Coupling of internal and surface waves in water of variable depth : Cox C. and H. Sandstrom, 1962. J. Oceanogr. Soc., Jap, 20th Ann. Vol.: 499–513

In this paper, a computationally efficient Monte Carlo simulation (MCS) approach is introduced in order to estimate the extreme response statistics of parametric rolling in random longitudinal seas. Basically, parametric roll is caused by sufficiently large oscillations of the roll restoring moment occurring within certain frequencies of wave encounter (approximately twice the natural roll frequency). The concept of the Grim effective wave is applied herein in order to approximate the variation of the restoring moment in random waves. A fourth order linear filter is introduced to model the random effective wave amplitude process, which is assumed to be the driving process for the variation of the restoring term as well as for the stochastic nonlinear system of the roll motion by application of the Grim effective wave approximation. For the stochastic dynamical system, the roll response is a random process and an extrapolation procedure is developed for estimating the extreme values of the response statistics by assuming regular behavior in the tail region of the mean upcrossing rate. The rationality of the linear filter model and the feasibility of an efficient MCS method based on extrapolation techniques for predicting the extreme roll response are illustrated. Furthermore, the phenomenon of parametric roll in random seas as well as the effect of vessel speed on the stochastic roll response are investigated.

SYNTHESIS OF OPTIMUM FIRST ORDER FILTERS FOR FLAT SPECTRA COMMAND AND NOISE SIGNALS∗

SUMMARY Command and noise signals expected to operate on specific control systems are often only scantily known at the design stage. This paper considers some of the problems which may arise in choosing a system transfer function when supplied with such inadequate information. A first order linear filter has been considered, and for synthesis purposes both command and noise signals have been represented by flat power spectra. The total power content and bandwidth of both signals have been varied, and the optimum transfer function for the various combinations determined by using the mean square error criterion. With the above assumptions, it is shown that under the circumstances most likely to be met in practice, and in the absence of more specific information, a time constant of one-quarter of the inverse of the expected command signal bandwidth will yield reasonable mean square error performance.