Linear Parameter Systems Research Articles

Matrix-valued Laurent polynomials, parametric linear systems and integrable systems

In this paper, we study transfer functions corresponding to parametric linear systems whose coefficients are block matrices. Thus, these transfer functions constitute Laurent polynomials whose coefficients are square matrices. We assume that block matrices defining the parametric linear systems are solutions of an integrable hierarchy called by us, the block matrices version of the finite discrete KP hierarchy, which is introduced and studied with certain detail in this paper. We see that the linear system defined of the simplest solution of the integrable system is controllable and observable. Then, as a consequence of this fact, it is possible to verify that any solution of the integrable hierarchy, obtained by the dressing method of the simplest solution, defines a parametric linear system which is also controllable and observable.

Adaptive Damping Design of PMSG Integrated Power System with Virtual Synchronous Generator Control

With the continuous development of wind power capacity, a large number of wind turbines connected by power electronic devices make the system inertia lower, which leads to the problem of system frequency stability degradation. The virtual synchronous generator (VSG) control can make wind turbines possess inertia and damping. However, the stochastic dynamic behavior of wind generation results in the stochastic changing of operating condition; this paper presents an adaptive subsynchronous oscillation (SSO) damping control method for the wind generation with VSG control. Firstly, the small signal model of the permanent magnet synchronous generator (PMSG) with VSG is built, and the model of state space is derived and built. The active power of PMSG is selected as the variable parameter vector to establish a polytopic linear variable parameter system model. Then, based on the hybrid H2/H∞ control method, each vertex state feedback matrix is solved by linear matrix inequality, and a subsynchronous oscillation adaptive damping controller with polytope is obtained. Finally, the 4-machine 2-area system connected to two PMSGs with VSG control is used as the test system for time domain simulation. The simulation results demonstrate that the LPV based adaptive damping controller could provide enough damping under the circumstances of wider changes of wind power outputs.

New parameterized solution with application to bounding secondary variables in FE models of structures

In this work we propose a new kind of parameterized outer estimate of the united solution set to an interval parametric linear system. The new method has several advantages compared to the methods obtaining parameterized solutions considered so far. Some properties of the new parameterized solution, compared to the parameterized solution considered so far, and a new application direction are presented and demonstrated by numerical examples. The new parameterized solution is a basis of a new approach for obtaining sharp bounds for derived quantities (e.g., forces or stresses) which are functions of the displacements (primary variables) in interval finite element models (IFEM) of mechanical structures.

Application of a novel linearization method to compare the on–off control strategies modeled by piecewise linear systems

In this study, a novel approach for linearization of piecewise linear systems is applied to approximate the root mean square output of a quarter car model with semiactive control. The on–off control strategies, balance logic, skyhook, groundhook, and hybrid are modeled by piecewise linear systems with variable friction. By the proposed method, one can attach, for each output of practical interest, a linear system with the same transmissibility factor. The obtained transmissibility factors are used to approximate the root mean square output of considered semiactive systems using the power spectral density input–output relationships for constant parameter linear systems with stationary random inputs. The method is applied for optimization of hybrid control strategy with respect to a performance index defined in terms of sprung mass acceleration, suspension travel, and dynamic contact force.

Virtual output feedback in parameter estimation

The use of poles as targets for estimating parameters of linear systems is attractive from a variance perspective but the limitations are well-known, namely: the number of poles that can be identified is often smaller than the parameters to be estimated and their sensitivity can be small. One way to mitigate these limitations, initially put forth about 20 years ago, is to collect data while the structure operates in closed-loop. In this manner, poles for different gains can be aggregated, and some control over the sensitivities can be realized by gain design. This paper shows that closed-loop interrogation can be realized without physical feedback by signal processing supported on the relation between open-loop and closed-loop transfer matrices for a given gain. Guidelines for closed-loop pole locations that provide robustness are derived and two numerical examples that exemplify the scheme in a model updating framework are included.

Dynamic characterization and control of a parallel haptic interaction with an admittance type virtual environment

Haptic interfaces, a kinesthetic link between a virtual environment and a human operator play a pivotal role in the reproduction of realistic haptic force feedback of the virtual reality-based simulators. Since most of the practical control theories are model-based, the identification of the robot’s dynamics, for precise modeling of the system dynamics, is a process of high significance and usage. This research addresses dynamic characterization, performance issues, and structural stability, associated with a parallel haptic device interaction with an admittance type virtual environment. In this regard, considering the Lion identification scheme, we characterized the dynamics of a robot nonlinearities in different operational points in the workspace, as a piecewise linear functions. Besides, utilizing the Lyapunov stability theory, we guaranteed the stability of the error dynamic of estimations. Next, by proposing a modified robust model predictive control scheme, one can guarantee the robust output feedback stability for a linear parametric variable system based on a quasi-min–max algorithm, subjected to the input constraints; besides, reducing the unwanted disturbances of the control efforts’ signals caused by switching in the piecewise linear dynamics. The simulations and the experimental implementation of a single-degree-of-freedom virtual-based haptic system are carried out. Results indicate that the identification is of high accuracy, and the appropriate convergence of parameters to the specific values are outlined. Besides, the output tracking error and its derivative behave desirably. The proposed method guarantees the robust stability of the system with output feedback subjected to input constraints. Uncertainties are well addressed in the prediction dynamics, thereby improving the output signals’ convergence to the desired output, being well suited for experimental practices of the haptic system.

Optimization of Linear Seismic Isolation Parameters via Crow Search Algorithm

In this study, isolation parameters of a seismic isolated structure were optimized via crow search algorithm (CSA). A shear frame model was generated with a base isolation system. The aim is to obtain isolation system parameters that result the minimum roof acceleration without exceeding the isolator displacement limits. The Matlab program was used for optimization operations and the SAP2000 program was used for structural analysis. The open application-programming interface (OAPI) was employed for the two programs to work together. Time history analyses were conducted using recorded near- and far-fault earthquake ground motions. Within the content of the study, linear isolator system parameters, which are the isolator period and damping ratio, were optimized through the linear time history analyses results by achieving the predefined aim. The analyses were performed for various isolation displacement limits and damping ratio ranges. The optimum results were determined at the range of various displacement limits and damping ratios by attaining the specified upper limits for displacement in all cases. Since higher damping ratios cause increased floor accelerations, optimum isolation parameters were achieved generally at the intermediate damping ratios well below the upper limits. Therefore, optimum isolation parameters are crucial for obtaining minimum floor accelerations. The verification of the optimum isolator parameters, which are achieved by the crow search algorithm employed, was made by comparing them with the results of a previous study.

Algebraic tests for the asymptotic stability of parametric linear systems

In this paper, algebraic tests are proposed to establish asymptotic stability of parametric linear systems assuming that the parameters lie in a given variety. In particular, by using tools borrowed from algebraic geometry, necessary and sufficient condition are proposed to test whether a continuous-time or a discrete-time linear parametric system is asymptotically stable for all the specializations belonging to a given variety. Both the cases of zero dimensional and non-zero dimensional varieties are considered.

Algebraic certificates for the structural properties of parametric linear systems

In this paper, by exploiting the concept of polynomial greatest common divisor, some algebraic tests are proposed to certify the structural properties of both discrete-time and continuous-time linear systems. Furthermore, by exploiting the concept of parametric greatest common divisor, such results are extended to certify the structural properties of systems whose dynamical matrices depend polynomially on some parameters.

Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness

In a paper by Willems et al., it was shown that persistently exciting data can be used to represent the input-output behavior of a linear system. Based on this fundamental result, we derive a parametrization of linear feedback systems that paves the way to solve important control problems using data-dependent linear matrix inequalities only. The result is remarkable in that no explicit system's matrices identification is required. The examples of control problems we solve include the state and output feedback stabilization, and the linear quadratic regulation problem. We also discuss robustness to noise-corrupted measurements and show how the approach can be used to stabilize unstable equilibria of nonlinear systems.

Optimum design of seismic isolation systems using metaheuristic search methods

This study addresses the optimum design of seismic isolated structures via metaheuristic search methods. Three recently developed bio-inspired search methods, namely crow search (CSA), whale optimization (WOA) and grey wolf optimizer (GWO), were employed to develop efficient design optimization algorithms for parameter optimization of seismic isolated structures. The developed design optimization algorithms have been applied to optimize a shear frame model with a base isolation system, where the main objective was to obtain isolation system parameters resulting in minimum roof acceleration without exceeding the isolation system displacement limits. The linear isolator system parameters, namely the damping ratio and isolator period, were optimized based on the time history analyses. The optimization procedure was performed for various isolation displacement limits and damping ratio ranges. The results obtained for this isolated structure by the three design optimization algorithms were compared with previously published results in the literature in order to achieve a verification of the developed algorithms. One of the main outcomes of this study is that the high damping ratio does not always guarantee optimum isolation parameters, while an optimum damping ratio is crucial for achieving the minimum roof acceleration.

Forecasting with Alternative Data

We consider the problem of forecasting fine-grained company financials, such as daily revenue, from two input types: noisy proxy signals a la alternative data (e.g. credit card transactions) and sparse ground-truth observations (e.g. quarterly earnings reports). We utilize a classical linear systems model to capture both the evolution of the hidden or latent state (e.g. daily revenue), as well as the proxy signal (e.g. credit cards transactions). The linear system model is particularly well suited here as data is extremely sparse (4 quarterly reports per year). In classical system identification, where the central theme is to learn parameters for such linear systems, unbiased and consistent estimation of parameters is not feasible: the likelihood is non-convex; and worse, the global optimum for maximum likelihood estimation is often non-unique. As the main contribution of this work, we provide a simple, consistent estimator of all parameters for the linear system model of interest; in addition the estimation is unbiased for some of the parameters. In effect, the additional sparse observations of aggregate hidden state (e.g. quarterly reports) enable system identification in our setup that is not feasible in general. For estimating and forecasting hidden state (actual earnings) using the noisy observations (daily credit card transactions), we utilize the learned linear model along with a natural adaptation of classical Kalman filtering (or Belief Propagation). This leads to optimal inference with respect to mean-squared error. Analytically, we argue that even though the underlying linear system may be "unstable," "uncontrollable," or "undetectable" in the classical setting, our setup and inference algorithm allow for estimation of hidden state with bounded error. Further, the estimation error of the algorithm monotonically decreases as the frequency of the sparse observations increases. This, seemingly intuitive insight contradicts the word on the Street. Finally, we utilize our framework to estimate quarterly earnings of 34 public companies using credit card transaction data. Our data-driven method convincingly outperforms the Wall Street consensus (analyst) estimates even though our method uses only credit card data as input, while the Wall Street consensus is based on various data sources including experts' input.

A New Error Estimator for Reduced-Order Modeling of Linear Parametric Systems

Efficient error estimation is important for reliable reduced-order modeling with guaranteed accuracy. We propose an error estimator for reduced-order modeling of linear parametric dynamical systems. The error estimator estimates the error of the reduced transfer function in the frequency domain and can be easily extended to the output error estimation of the reduced-order models (ROMs) for linear steady parametric systems. It is tight and cheap to compute. Using the error estimator, the ROM can be adaptively obtained with high reliability. Numerical results show that the error estimator can accurately estimate the true error even for transfer functions with many resonances. Compared with an existing error bound, the proposed error estimator can be orders of magnitudes sharper and needs much less computational time.

Identification of Structural Parameters and Unknown Inputs Based on Revised Observation Equation: Approach and Validation

The identification of parameters of linear or nonlinear systems under unknown inputs and limited outputs is an important but still challenging topic in the context of structural health monitoring. Time-domain analysis methodologies, such as extend Kalman filter (EKF), have been actively studied and shown to be powerful for parameter identification. However, the conventional EKF is not applicable when the input is unknown or unmeasured. In this paper, by introducing a projection matrix in the observation equation, a time-domain EKF-based approach is proposed for the simultaneous identification of structural parameters and the unknown excitations with limited outputs. A revised version of observation equation is presented. The unknown inputs are identified using the least squares estimation based on the limited observations and the estimated structural parameters at the current time step. Particularly, an analytical recursive solution is derived. The accuracy and effectiveness of the proposed approach is first demonstrated via several numerical examples. Then it was validated by the shaking table tests on a five-story building model for the robustness in application to real structures. The results show that the proposed approach can satisfactorily identify the parameters of linear or nonlinear structures under unknown inputs.

Analysis of linear lung models based on state-space models

Background and Objectives: Linear parametric respiratory system models have been used in the model-based analysis of the respiratory system. Although there are studies exploring the physiological correctness and fitting accuracy of the models, they are not analysed in terms of interaction between parameters and dynamics of the model. In this study we propose to use state-space modelling to yield the time-varying nature of the system incorporated by the parameters. Methods: We tested controllability, observability and stability characteristics of the equation of motion, 2-comp. parallel, 2-comp. series, viscoelastic, 6-element and mead models while using the parameters given in the literature. In the sensitivity analysis we proposed to use dual Desensitized Linear Kalman Filter (DKF) and Extended Kalman Filter (EKF) method. In this method, state error covariance revealed the parameter sensitivities for each model. Results: Results showed that all models, except 2-comp. parallel and mead models, are both controllable and observable models. On the other hand all models, except mead model, are stable models. Regarding to the sensitivity analysis, dual DKF - EKF method estimated states of the models successfully with a low estimation error. Sensitivity analysis results showed that airway parameters have higher effects on the state estimation than the other parameters have. Conclusion: We proved that state-space evaluation of the previously proposed parametric models of the respiratory system led us to quantitative and qualitative assessments of the respiratory models. Moreover parameter values found in the literature have different effects on the models.

Differential Galois Theory and Isomonodromic Deformations

We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group $G$ is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.

Enclosing the solution set of the parametric generalised Sylvester matrix equation A(p)XB(p) + C(p)XD(p) = F(p)

In this paper, we investigate the parametric generalised Sylvester matrix equation , whose elements are linear functions of uncertain parameters varying within intervals. This model generalises both interval matrix equations and parametric interval linear systems, so it is a quite general model. First, we give some sufficient conditions under which the solution set of this parametric equation is bounded. We then propose several approaches for enclosing the solution set that acquire tighter enclosures than those obtained by relaxing the parametric system to an interval system. Some special cases of interval systems that inherently are parametric (have dependent structure) are considered, too. Finally, numerical experiments are given to illustrate the effectiveness of the proposed methods.

Benchmark problems for continuous-time model identification: Design aspects, results and perspectives

The problem of estimating continuous-time model parameters of linear dynamical systems using sampled time-domain input and output data has received considerable attention over the past decades and has been approached by various methods. The research topic also bears practical importance due to both its close relation to first principles modelling and equally to linear model-based control design techniques, most of them carried in continuous time. Nonetheless, as the performance of the existing algorithms for continuous-time model identification has seldom been assessed and, as thus far, it has not been considered in a comprehensive study, this practical potential of existing methods remains highly questionable. The goal of this brief paper is to bring forward a first study on this issue and to factually highlight the main aspects of interest. As such, an analysis is performed on a benchmark designed to be consistent both from a system identification viewpoint and from a control-theoretic one. It is concluded that robust initialization aspects require further research focus towards reliable algorithm development.

Generalized Störmer and dynamical behavior of charged particles near magnetic planet

We study the motion of charged dust grains at arbitrary distants from a rotating magnetic planet,we show that the model treated for dipole field is an excellent approximation in most cases for a generalized Stormer problem, But that fundamentally new physics arises with the inclusion of a quadrupolar magnetic term. In particular,the appearence of a new trajectories for different kind of charged dust grains,which are meaningful when the individual and a comparable impact of the co-rotational and gravitational fields is considered.Furthermore,a detailed mathematical treatment of equilibrium state for various phase spaces for charged components in prograde or retrograde rotation direction with analytical solutions for a non linear parametric system.

A New Cluster-Based Harmonic Balance Aided Optimization Procedure With Application to Nonlinear Vibration Absorbers

This work presents a new strategy in design optimization of nonlinear vibration absorbers with continuous and discontinuous motions. A cluster-based harmonic balance aided optimization technique using force balance or energy balance as the basis is generalized and adapted for nonlinear systems. It is found that optimal design parameters form a cluster in the parameter space and points from the parameter space inside the cluster satisfies design considerations. One of the main disadvantages of using existing optimization methods in nonlinear systems is that the parameter regimes, which provide periodic solutions, are not known beforehand, so one has to first do bifurcation studies to arrive at periodic regimes and optimization has to be conducted in the range. Proposed method combines these two steps as it converges to periodic clusters alone. Since the method admits only periodic solutions, occurrence of conditions such as chaos and quasi periodicity can be eliminated from the dynamics of the system. The proposed method can also be used to find the optimal parameters of both linear and nonlinear dynamical systems.