Single Linear System Research Articles

From acceleration-based semi-active vibration reduction control to functional observer design

Abstract This work investigates a functional estimation problem for single input single output linear and nonlinear systems, motivated by its enabling role in acceleration-based semi-active control. Solvability of a linear functional estimation problem is studied from a geometric approach, where the functional dynamics are derived, decomposed, and transformed to expose structural properties. This approach is extended to solve a challenging nonlinear functional observer problem, combining with the exact error linearization. Existence conditions of nonlinear functional observers are established. Simulation verifies existence conditions and demonstrates the effectiveness of the proposed functional observer designs.

On model order priors for Bayesian identification of SISO linear systems

ABSTRACTA method for the identification of single input single output linear systems is presented. The method employs a Bayesian approach to compute the posterior distribution of the model parameters given the data-set. Since this distribution is often unavailable in closed form, a Metropolis Hastings algorithm is implemented to draw samples from it. To implement the sampler, the inclusion of prior information regarding the model order of the identified system is discussed. As one of the main contributions of this work, a prior over the Hankel singular values of the model is imposed. Numerical examples illustrate the method.

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

Abstract We consider settings for which one needs to perform multiple flow simulations based on the Navier–Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

Input–Output Finite Time Stabilization of Time-Varying Impulsive Positive Hybrid Systems under MDADT

Time-varying impulsive positive hybrid systems based on finite state machines (FSMs) are considered in this paper, and the concept of input–output finite time stability (IO-FTS) is extended for this type of hybrid system. The IO-FTS analysis of the single linear time-varying system is given first. Then, the sufficient conditions of IO-FTS for hybrid systems are proposed via the mode-dependent average dwell time (MDADT) technique. Moreover, the output feedback controller which can stabilize the non-autonomous hybrid systems is derived, and the obtained results are presented in a linear programming form. Finally, a numerical example is provided to show the theoretical results.

Monolithic coupling of the pressure and rigid body motion equations in computational marine hydrodynamics

In Fluid Structure Interaction (FSI) problems encoun- tered in marine hydrodynamics, the pressure field and the velocity of the rigid body are tightly coupled. This coupling is traditionally resolved in a partitioned manner by solving the rigid body motion equations once per nonlinear correction loop, updating the position of the body and solving the fluid flow equations in the new configura- tion. The partitioned approach requires a large number of nonlinear iteration loops per time–step. In order to enhance the coupling, a monolithic approach is proposed in Finite Volume (FV) framework, where the pressure equation and the rigid body motion equations are solved in a single linear system. The coupling is resolved by solving the rigid body motion equations once per linear solver iteration of the pressure equation, where updated pressure field is used to calculate new forces acting on the body, and by introducing the updated rigid body boundary velocity in to the pressure equation. In this paper the monolithic coupling is validated on a simple 2D heave decay case. Additionally, the method is compared to the traditional partitioned approach (i.e. “strongly coupled” approach) in terms of computational efficiency and accuracy. The comparison is performed on a seakeeping case in regular head waves, and it shows that the monolithic approach achieves similar accuracy with fewer nonlinear correctors per time–step. Hence, significant savings in computational time can be achieved while retaining the same level of accuracy.

Adaptive Semi-Implicit Integrator for Articulated Mechanical Systems

This paper concerns the dynamic simulation of constrained mechanical systems in the context of real-time applications and stable integrators. The goal is to adaptively find a balance between the stability of an over-damped implicit scheme and the energetic consistency of the symplectic, semi-implicit Euler scheme. As a starting point, we investigate in detail the properties of a recently proposed timestepping scheme, which approximates a full nonlinear implicit solution with a single linear system, without compromising stability. This scheme introduces a geometric stiffness term that improves numerical stability up to a certain time-step size, but it does so at the cost of large mechanical dissipation in comparison to the traditional constrained dynamics formulation. Dissipation is sometimes undesirable from a mechanical point of view, especially if the dissipation is not quantified. In this paper, we propose to use an additional control parameter to regulate “how implicit” the Jacobian matrix is, and change the degree to which the geometric stiffness term contributes. For the selection of this parameter, adaptive schemes are proposed based on the monitoring of energy drift. The proposed adaptive method is verified through the simulation of open-chain systems.

Implementation of Proportional-Integral-Observer Techniques for Load Frequency Control of Power System

Abstract: This paper presents the implementation of Proportional-Integral-Observer(PI-Observer) based state feedback control for Load Frequency of single area isolated power system model. The general application of PI-Observer has been realized in areas of robotic and mechanical system. The PI-Observer is used to estimate both the original states and the unknown inputs generally disturbances, modeling errors or the other non-linearities present in the system. The estimation error of the PI-Observer is gradually reduced by the proportional and integral feedback loop that appreciably improves the observer dynamics. A single area linear power system model is taken as a test case to implement the proposed methodology which is then compared with full state Luenberger observer based state feedback control for load frequency control of power system. MATLAB/SIMULINK is extensively used for modeling & simulation of power system and observer. The dynamic responses are compared and discussed on the basis of error dynamics, system states, peak overshoot and settling time. Lastly it was found that response of PI-observer based state feedback control scheme is quite appreciable as compared to Luenberger observer based state feedback schemes.

Adaptive Input Design for LTI Systems

Optimal input design for parameter estimation has obtained extensive coverage in the past. A key problem here is that the optimal input depends on some unknown system parameters that are to be identified. Adaptive design is one of the fundamental routes to handle this problem. Although there exist a rich collection of results on this problem, there are few results that address dynamical systems. This paper presents sufficient conditions for convergence/consistency and asymptotic optimality for a class of adaptive systems consisting of a recursive prediction error estimator and an input generator depending on the time-varying parameter estimates. The results apply to a general family of single input single output linear time-invariant systems. An important application is adaptive input design for which the results imply that, asymptotically in the sample size, the adaptive scheme recovers the same accuracy as the off-line prediction error method that uses data from an experiment where perfect knowledge of the system has been used to design an optimal input spectrum.

Semi-implicit anisotropic cosmic ray transport on an unstructured moving mesh

In the interstellar medium of galaxies and the intracluster gas of galaxy clusters, the charged particles making up cosmic rays are moving almost exclusively along (but not across) magnetic field lines. The resulting anisotropic transport of cosmic rays in the form of diffusion or streaming not only affects the gas dynamics but also rearranges the magnetic fields themselves. The coupled dynamics of magnetic fields and cosmic rays can thus impact the formation and evolution of galaxies and the thermal evolution of galaxy clusters in critical ways. Numerically studying these effects requires solvers for anisotropic diffusion that are accurate, efficient, and robust, requirements that have proven difficult to satisfy in practice. Here, we present an anisotropic diffusion solver on an unstructured moving mesh that is conservative, does not violate the entropy condition, allows for semi-implicit time integration with individual timesteps, and only requires solving a single linear system of equations per timestep. We apply our new scheme to a large number of test problems and show that it works as well or better than previous implementations. Finally, we demonstrate for a numerically demanding simulation of the formation of an isolated disk galaxy that our local time-stepping scheme reproduces the results obtained with global time-stepping at a fraction of the computational cost.

Evaluation of the validity of the scalar approximation in optical wave propagation using a systems approach and an accurate digital electromagnetic model

The cause and amount of error arising from the use of the scalar approximation in monochromatic optical wave propagation are discussed using a signals and systems formulation. Based on Gauss’s Law, the longitudinal component of an electric field is computed from the transverse components by passing the latter through a two input single output linear shift-invariant system. The system is analytically characterized both in the space and frequency domains. For propagating waves, the large response for the frequencies near the limiting wave number indicates the small angle requirement for the validity of the scalar approximation. Also, a discrete simulator is developed to compute the longitudinal component from the transverse components for monochromatic propagating electric fields. The simulator output helps to evaluate the validity of the scalar approximation when the system output cannot be analytically calculated.

On the Input-Output Distinguishability of Single Output Continuous Linear Time-Invariant Systems

This note addresses the distinguishability problem for continuous linear time-invariant systems using their input and output data. The proposed approach is based on the characterization of the class of initial state vectors and control input signals that make the outputs of different continuous linear time-invariant systems indistinguishable. This class is defined here as the “indistinguishability zone.” From this characterization, a simple necessary and sufficient condition for the distinguishability of single-input, single-output systems is established. Furthermore, it is shown that multiple-input, single-output systems are never distinguishable.

Deterministic analysis of distributed order systems using operational matrix

Recently, distributed order systems as a generalized concept of fractional order have been a major focus in science and engineering areas, and have rapidly extended application across a wide range of disciplines. However, only a few numerical methods are available for analyzing the distributed order systems. This paper proposes a novel numerical scheme to analyze the behavior of single input single output linear systems in the time domain with a single distributed order differentiator/integrator by using operational matrix technique. The proposed method reduces different analysis problems to a system of algebraic equations by using block pulse functions, which makes it easy to handle an arbitrary input. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method. The proposed method was found to be an efficient tool for analyzing linear distributed order systems.

Charge and current beats in T-shaped qubit–detector systems

The time evolution of a charge qubit coupled electrostatically with different detectors in the forms of single, double and triple quantum dot linear systems in the T-shaped configuration between two reservoirs is theoretically considered. The correspondence between the qubit quantum dot oscillations and the detector current is studied for different values of the inter-dot tunneling amplitudes and the qubit–detector interaction strength. We have found that even for a qubit coupled with a single QD detector, the coherent beat patterns appear in the oscillations of the qubit charge. This effect is more evident for a qubit coupled with double or triple-QD detectors. The beats can be also observed in both the detector current and the detector quantum dot occupations. Moreover, in the presence of beats the qubit oscillations hold longer in time in comparison with the beats-free systems with monotonously decaying oscillations. The dependence of the qubit dynamics on different initial occupations of the detector sites (memory effect) is also analyzed.

Multirate parallel distributed compensation of a cluster in wireless sensor and actor networks

The stabilisation problem for one of the clusters with bounded multiple random time delays and packet dropouts in wireless sensor and actor networks is investigated in this paper. A new multirate switching model is constructed to describe the feature of this single input multiple output linear system. According to the difficulty of controller design under multi-constraints in multirate switching model, this model can be converted to a Takagi–Sugeno fuzzy model. By designing a multirate parallel distributed compensation, a sufficient condition is established to ensure this closed-loop fuzzy control system to be globally exponentially stable. The solution of the multirate parallel distributed compensation gains can be obtained by solving an auxiliary convex optimisation problem. Finally, two numerical examples are given to show, compared with solving switching controller, multirate parallel distributed compensation can be obtained easily. Furthermore, it has stronger robust stability than arbitrary switching controller and single-rate parallel distributed compensation under the same conditions.

PWLM[formula omitted]-based automatic performance model estimation method for HDFS write and read operations

There is a growing need for the development of an automatic performance model estimation method for Hadoop Distributed File System (HDFS) write and read (W/R) operations in order to deal with constant software improvement and updates, parameter configuration changes, hardware heterogeneity, and their Quality of Service (QoS) evaluation. Extant research based on single linear system model has a limited ability to explain the performance variations due to changes in HDFS parameters such as block size. These variations reveal some typical characteristics of nonlinear systems and are an obstacle in achieving effective automatic performance estimation. In order to deal with this challenge, a piecewise-linear multi-model modeling (PWLM3)-based automatic performance model estimation method is proposed for HDFS W/R performance. In the proposed method, a standard model base is built to standardize the model representation of every submodel. Moreover, a cluster quality assessment strategy is applied to evaluate the optimal number of submodels, and a submodel selection strategy is implemented to construct performance model candidates and improve the computation efficiency of the proposed method. In addition, Levenberg–Marquardt (LM) and Universal Global Optimization (UGO) algorithms are adopted to estimate the values of switch points and identify undetermined parameters of performance model candidates. Then the performance model is selected among these candidates according to Root Mean Squared Error (RMSE) indicator. Experimental results demonstrate that the PWLM3-based performance model provides a good understanding and description of nonlinear characteristics of HDFS W/R performance and achieves better identification precision than a single linear system model-based one.

Time Optimal Feedback Control using Chebyshev Polynomials and Gröbner Basis

The problem of synthesizing time optimal “feedback” control for a single input linear time invariant system with constrained control input is considered. It is well known that the optimal open loop control is bang-bang and the switching takes place according to some switching surfaces in the state space. A parametric representation of the switching surfaces using Pontryagins’ minimum principle (PMP) is first computed. This parametric representation is in general not semi-algebraic. For such situations, Chebyshev polynomials are used to approximate the switching surfaces. Then, a Gröbner basis based elimination algorithm is used to find an implicit representation for approximated switching surfaces. The computed representation is used to construct a state feedback based nested switching logic which will achieve a near time optimal state transfer of system states to the origin.

Order reduction using improved pole clustering with Pade approximation method for linear system

An enhanced pole clustering method is presented that yields linear time invariant-reduced order system. This is accomplished by using the combined benefits of improved Pade approximation method. The applicability of the proposed method for both single and multi-variable linear time invariant systems is verified by solving numerical experiment. Furthermore, the results show that the realised LTI-reduced order system retains stability and steady-state conditions of the original system. The reduced model also exhibits lower integral square error values.

三峡水库入库洪水分析计算研究

三峡水库建成蓄水后，水库周边区间流域产汇流条件发生改变，新形成的入库洪水较天然条件下的坝址洪水具有“洪峰增大，涨水段洪量增大”等特性。本文基于多输入单输出系统模型模拟计算三峡建库前的入库流量系列，定量分析了三峡入库与坝址的年最大洪峰、3、7、15 d洪量之间的关系，整编得到三峡入库流量不连续序列，据此开展三峡入库洪水频率分析，并与坝址计算成果进行比较。结果表明，入库系列与坝址系列的洪峰以及3 d、7 d洪量差异明显，15 d洪量变化不大，入库百年一遇设计洪峰较坝址设计洪峰增大7.89%，入库百年一遇3 d、7 d、15 d洪量较坝址设计洪量分别增大5.49%、2.52%和0.31%，现状条件下采用入库洪水设计值作为三峡水库防洪调度的科学依据更为合理。 The runoff generation and confluence conditions at intervener basin have been changed after the com-pletion of the Three Gorges Reservoir (TGR). Since the reservoir inflow has higher peak with larger flood volume in the rising limb compared to the floods at dam site, the original designed flood results at dam site can no longer satisfy the current requirement of operation and management for TGR. In this paper, a multiple inputs and single output linear system model was built to calculate the historical reservoir inflow series and then linear regressions were used to determine the quantitative relations of flood peaks, 3 d, 7 d and 15 d flood volumes between inflow series and flood series at dam site. Subsequently, the simulated inflow series combined with historical floods were obtained and used for inflow frequency analysis. Comparing the inflow frequency analysis results with the existing design flood at dam site, the flood peaks and 3d, 7 d flood volumes are increased significantly, while the change of 15 d flood volume is not evident. It is shown that 100-year designed inflow peak discharge, 3 d, 7 d and 15 d inflow volumes are 7.89%, 5.49%, 2.52% and 0.31% larger than those of designed floods at dam site, respectively. Therefore, it’s more rational to take the designed inflow floods as scientific bases for TGR reservoir operation and management.

Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming non-symmetric matrices for both recycling BiCG and recycling BiCGSTAB). As compared with other methods for solving sequences of single linear systems with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based recycling algorithms, like recycling BiCGSTAB, have the advantage that they involve a short-term recurrence, and hence, do not suffer from storage issues and are also cheaper with respect to the orthogonalizations. We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametric model order reduction example gives good results. We show about 40% savings in the number of matrix-vector products and about 35% savings in runtime.

BiCR-type methods for families of shifted linear systems

The shifted linear systems with non-Hermitian matrices often arise from the numerical solutions for time-dependent PDEs, computing the large-scale eigenvalue problems, control theory and so on. In the present paper, we develop two shifted variants of BiCR-type methods for solving such linear systems. These variants of BiCR-type methods take advantage of the shifted structure, so that the number of matrix–vector multiplications and the number of inner products are the same as a single linear system. Finally, extensive numerical examples are reported to illustrate the performance and effectiveness of the proposed methods.