Linear Dynamic System Model Research Articles

Networked Assembly of Mechatronic Linear Physical System Models

Engineering design is evolving into a global activity. Globally distributed design requires efficient global distribution of models of dynamic physical systems through computer networks. These models must describe the external input-output behavior of the electrical, mechanical, fluid, and thermal dynamics of engineering systems. An efficient system model assembly method is then required to assemble these component system models into a model of a yet higher-level dynamic system. Done recursively, these higher-level system models become possible components for yet higher-level analytical models composed of external model equations in the same standardized format as that of the lowest level components. Real-time, automated exchange, and assembly of engineering dynamic models over a global network requires four characteristics. The models exchanged must have a unique standard format so that they can be exchanged and assembled by an automated process. The exchange of model information must be executed in a single-query transmission to minimize network load. The models must describe only external behavior to protect internal model details. Finally, the assembly process must be recursive so that the transfer and assembly processes do not change with the level of the model exchanged or assembled. This paper will introduce the modular modeling method (MMM), a modeling strategy that satisfies these requirements. The MMM distributes and assembles linear dynamic physical system models with a dynamic matrix representation. Using the MMM method, dynamic models of complex assemblies can be built and distributed while hiding the topology and characteristics of their dynamic subassemblies.

Active Estimation for Jump Markov Linear Systems

Jump Markov Linear Systems are convenient models for systems that exhibit both continuous dynamics and discrete mode changes. Estimating the hybrid discrete-continuous state of these systems is important for control and fault detection. Existing solutions for hybrid estimation approximate the belief state by maintaining a subset of the possible discrete mode sequences. This approximation can cause the estimator to lose track of the true mode sequence when the effects of discrete mode changes are subtle. In this paper, we present a method for active hybrid estimation, where control inputs can be designed to discriminate between possible mode sequences. By probing the system for the purposes of estimation, such a sequence of control inputs can greatly reduce the probability of losing the true mode sequence compared to a nominal control sequence. Furthermore, by using a constrained finite horizon optimization formulation, we are able to guarantee that a given control task is achieved, while optimally detecting the hybrid state. In order to achieve this, we present three main contributions. First, we develop a method by which a sequence of control inputs is designed in order to discriminate optimally between a finite number of linear dynamic system models. These control inputs minimize a novel, tractable upper bound on the probability of model selection error. Second, we extend this approach to develop an active estimation method for Jump Markov Linear Systems by relating the probability of model selection error to the probability of losing the true mode sequence. Finally, we make this method tractable using a principled pruning technique. Simulation results show that the new method applied to an aircraft fault detection problem significantly decreases the probability of a hybrid estimator losing the true mode sequence.

Robust Stability Analysis for Autonomous Min-Max Systems

Min-max systems are natural extensions of the well known (max,plus) linear discrete event dynamic system (DEDS) models. They are motivated by recent research on digital circuits and communication networks. In this paper, we study the robust stability of a family of min-max systems whose parameters are perturbed in intervals. In the spirit of Kharitonov, we first reduce the robust stability test to the check of stability on the vertex set of parameter space. We then further reduce the test to a subset derived from the structure of the system family.

Predicting state transitions in the transcriptome and metabolome using a linear dynamical system model.

BackgroundModelling of time series data should not be an approximation of input data profiles, but rather be able to detect and evaluate dynamical changes in the time series data. Objective criteria that can be used to evaluate dynamical changes in data are therefore important to filter experimental noise and to enable extraction of unexpected, biologically important information.ResultsHere we demonstrate the effectiveness of a Markov model, named the Linear Dynamical System, to simulate the dynamics of a transcript or metabolite time series, and propose a probabilistic index that enables detection of time-sensitive changes. This method was applied to time series datasets from Bacillus subtilis and Arabidopsis thaliana grown under stress conditions; in the former, only gene expression was studied, whereas in the latter, both gene expression and metabolite accumulation. Our method not only identified well-known changes in gene expression and metabolite accumulation, but also detected novel changes that are likely to be responsible for each stress response condition.ConclusionThis general approach can be applied to any time-series data profile from which one wishes to identify elements responsible for state transitions, such as rapid environmental adaptation by an organism.

Learning and Inferring Motion Patterns using Parametric Segmental Switching Linear Dynamic Systems

Switching Linear Dynamic System (SLDS) models are a popular technique for modeling complex nonlinear dynamic systems. An SLDS can describe complex temporal patterns more concisely and accurately than an HMM by using continuous hidden states. However, the use of SLDS models in practical applications is challenging for three reasons. First, exact inference in SLDS models is computationally intractable. Second, the geometric duration model induced in standard SLDSs limits their representational power. Third, standard SLDSs do not provide a principled way to interpret systematic variations governed by higher order parameters. The contributions in this paper address all of these three challenges. First, we present a data-driven MCMC (DD-MCMC) sampling method for approximate inference in SLDSs. We show DD-MCMC provides an efficient method for estimation and learning in SLDS models. Second, we present segmental SLDSs (S-SLDS), where the geometric distributions of the switching state durations are replaced with arbitrary duration models. Third, we extend the standard SLDS model with additional global parameters that can capture systematic temporal and spatial variations. The resulting parametric SLDS model (P-SLDS) uses EM to robustly interpret parametrized motions by incorporating additional global parameters that underly systematic variations of the overall motion. The overall development of the extensions for SLDSs provide a principled framework to interpret complex motions. The framework is applied to the honey bee dance interpretation task in the context of the on-going BioTracking project at the Georgia Institute of Technology. The experimental results suggest that the enhanced models provide an effective framework for a wide range of motion analysis applications.

Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems

This paper discusses a class of unexpected irreversible phenomena that can develop in linear conservative systems and provides a theoretical foundation that explains the underlying principles. Recent studies have shown that energy can be introduced to a linear system with near irreversibility, or energy within a system can migrate to a subsystem nearly irreversibly, even in the absence of dissipation, provided that the system has a particular natural frequency distribution. The present work introduces a general theory that provides a mathematical foundation and a physical explanation for the near irreversibility phenomena observed and reported in previous publications. Inspired by the properties of probability distribution functions, the general formulation developed here is based on particular properties of harmonic series, which form the common basis of linear dynamic system models. The results demonstrate the existence of a special class of linear nondissipative dynamic systems that exhibit nearly irreversible energy exchange and possess a decaying impulse response. In addition to uncovering a new class of dynamic system properties, the results have far-reaching implications in engineering applications where classical vibration damping or absorption techniques may not be effective. Furthermore, the results also support the notion of nearly irreversible energy transfer in conservative linear systems, which until now has been a concept associated exclusively with nonlinear systems.

Multiscale theory for linear dynamic processes: Part 1. Foundations

Multiscale models of linear dynamic systems offer a representation that links states at different time scales over distinct time periods. As such they allow a natural bridging of scales (ranges of frequencies) and time, and are more attractive than models defined in the time- or frequency-domain, alone. The term “multiscale model”, as used in this two-part series, is quite different from the notion of a “multiscale model”, as this is used in recent publications, where in essence such model bridges the states of a system at two distinct scales (an atomistic scale of fast elementary phenomena, and a macroscopic scale characterizing slow continuum-based physics), described by two distinctly different physical models. The multiscale dynamic models, introduced in this paper, link state dynamics over a number of distinct time scales, and are in essence “multiresolution” facets of a single model. The outgrowth of a series of developments, which came about with the advent of the wavelet decomposition for the analysis of discrete signals, multiscale models are defined on dyadic trees, which span the time-scale space. The nodes of such trees are used to index the values of states, inputs and outputs, modeling errors, and measurement errors of dynamic systems. These values are localized in both time and scale (range of frequencies), and thus they offer a hybrid domain that is particularly conducive for estimation and control problems. In Part 1 of this series, which focuses on the foundations of a multiscale systems theory, we will describe the generation of multiscale models from their time-domain counterparts, and develop conditions governing their stability, controllability and observability. In addition, we will address the development of algorithms for the solution of basic problems, such as: optimal control, state estimation, and optimal fusion of measurements and control actions at various scales. It will be shown that multiscale models lend themselves nicely to construction of very effective parallelizable algorithms. In Part 2 of this series, we will apply all these developments to the formulation of a Multiscale Model Predictive Control (MS-MPC) approach and we will show that MS-MPC is a more attractive framework for the design and implementation of multivariable control systems.

Fundamentals of apparent damping phenomena in linear conservative systems

This presentation addresses a class of irreversible phenomena that can develop in linear conservative systems and provides a theoretical foundation to explain the underlying principles. Recent studies have shown that energy can be introduced to a linear system with near irreversibility, or energy within a system can migrate to a subsystem nearly irreversibly, even in the absence of dissipation. Inspired by the properties of probability distribution functions, the general formulation developed here is based on particular properties of harmonic series, which form the common basis of linear dynamic system models. The results demonstrate the existence of a special class of linear nondissipative dynamic systems that exhibit nearly irreversible energy exchange and possess a decaying impulse response. The formulation and its results also support the recent studies that observed near irreversibility and apparent damping in several dynamic systems and provide a common theoretical foundation for such behavior. [Research carried out while AA served at NSF.]

Modeling Sensorimotor Learning with Linear Dynamical Systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation-the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a substantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.

Modeling Sensorimotor Learning with Linear Dynamical Systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation— the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a sub-stantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.

Instantaneous Optimal Control of Seismically-Excited Structures Using a Maximum Principle

An optimal control scheme is proposed for linear dynamic system models, and the optimal control forces are derived using a maximum principle. Numerical examples are presented demonstrating the characteristics of the control scheme. An instantaneous extension to the optimal control method is proposed to deal with the unknown nature of seismic disturbances. Examples of the effectiveness of the method and a tradeoff curve are presented.

Frequency Domain System Identification Toolbox for MATLAB: Automatic Processing - from Data to Models

The Frequency Domain System Identification Toolbox for MATLAB is an effective tool for the identification of linear dynamic system models from measured data. Since the use of advanced system identification methods often requires a lot of programming work, the attention can be deflected from the modelling issues. Therefore, a Graphical User Interface (GUI) was developed which allows the experimenter to visually follow and control the data processing and modelling steps.However, for many users it is still desirable to obtain good models with as few decisions to be made as it is possible. Therefore, automatic processing steps have been added to the GUI. Identification can be done now in this toolbox with a minimum of user interactions in the graphics windows, and a reasonable model is returned, ready for control or physical analysis.

A mixed-level switching dynamic system for continuous speech recognition

A two-level mixture linear dynamic system model, with frame-level switching parameters in the observation equation and with segment-level switching parameters in the target-directed state equation, is developed and evaluated. The main contributions of this work are: (1) the new framework for dealing with mixed-level switching in the dynamic system and (2) the novel use of piecewise linear functions, enabled by the introduction of frame-level switching, to approximate the nonlinear function between the hidden vocal-tract-resonance space and the observable acoustic space. The approximation is accomplished by the frame-dependent switching parameters in the observation equation. In this paper, in a self-contained manner, we highlight the key algorithm differences from the earlier model having only single segment-level switching that is synchronous between the state and observation equations. A series of speech recognition experiments are carried out to evaluate this new model using a subset of Switchboard conversational speech data. The experimental results show that the approximation accuracy is improved with an increased number of switching-parameter values. The speech recognizer built from the new mixed-level switching dynamic system model using an N-best re-scoring evaluation paradigm show moderate word error rate reduction compared with using either single-level switching or no switching parameters.

Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems

In recent years, a great deal of attention has been devoted to Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. The surge of interest was triggered by the pressing need for efficient numerical techniques for simulations of extremely large-scale dynamical systems arising from circuit simulation, structural dynamics, and microelectromechanical systems. In this paper, we begin with a tutorial of a Lanczos process based Krylov subspace technique for reduced-order modeling of linear dynamical systems, and then give an overview of the recent progress in other Krylov subspace techniques for a variety of dynamical systems, including second-order and nonlinear systems. Case studies arising from circuit simulation, structural dynamics and microelectromechanical systems are presented.

Putting the process in developmental processes

Several signs point to a strengthening of our capabilities for rigorously modelling developmental processes and other kinds of changes. The indicators of progress range from stronger formulations of “systems thinking” and definitions through measurement and design considerations to advanced mathematical representations, such as linear and nonlinear dynamical systems models. We believe these advances offer major improvements for the treatment of process and related concepts as they have evolved thus far largely within the meta-model of stability and equilibrium that has dominated much of science (Holling, 1973). These issues are summarised and some of the promising innovations that we believe will make the coming decades highly productive ones for the study of development will be discussed.

Orthonormal basis functions models: A transformation analysis

This paper presents a framework to analyze rational orthononiial basis functions for modeling and identification of discrete time linear dynamical systems. By using properties of orthogonal all-pass filters and multi-linear mappings, the transformation analysis of Laguerre models is extended to general Orthononiial Basis Functions (OBF) models. The basic idea is to map the system into a new domain, so that the energy of the impulse response of the transformed system is concentrated to its initial phase. Hence, the transformed system can be well approximated by a low order finite impulse response model. This approximate model is then transformed back to the original domain. This leads to an OBF model, which has infinite impulse response and is linear in the unknown parameters. Hence, such a model can be used in linear regression least squares system identification. Powerful tools for analyzing the properties of the corresponding transfer function estimates are provided.

Determining the stability and realizability of a type of identified power system linear dynamic equivalent model

Abstract The linear multi-input multi-output dynamic equivalent system model for power system studies is often required to be stable and realizable. The stability and realizability may be checked after the continuous state space model has been obtained, but it will result in excessive computing time for model identification. This paper gives the conditions on the poles of an autoregression model for checking its stability and realizability. Along with the assumption of output decoupling, application of these conditions can greatly reduce the computing time for model identification.

Interactive software for dynamic system modeling using linear graphic

A description is given of Lgraph, which is an interactive, graphical software package for creating, modifying, and reducing linear graphs. Linear graphing, a structured method for formulating and representing lumped-element, linear dynamic system models containing connected single-port energy storage, dissipative, and source elements, and two-port transduction elements, is explained through the example of a simplified, quarter-body car suspension. Lgraph enables the user to place elements from several domains in a connected graph, and then computes and displays elemental equations, constraint (loop and node) equations, and state equations for the dynamic system, all in symbolic form. Its operation is described by a second example, the modeling of a DC servomotor drive system. The procedural algorithms are described. The package was implemented using X Windows graphics and runs on Unix-based engineering workstations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Recursive transformation matrices for linear dynamic system models

In time series analyses and forecasting, dynamic linear models of canonical form are quite often used, specially in Generalised Exponentially Weighted Regression (GEWR) type models, introduced by Harrison and Akram (1982) and Akram (1984). This form is convenient, but usually not attractive for operation since the meaning of the parameters is not clear. To overcome this problem one needs a system of dynamic linear models in diagonal or any other meaningful form similar to canonical form. This is obtained by reparameterisation of a model to a desired form by similarity transformation of one system to another system through a transformation matrix. A general analytical expression for the elements of this matrix and its inverse, specially in case of canonical to diagonal transformation, is not known. One needs to compute these elements each time a transformation is sought. To ease this cumbersome problem a general recursive form of transformation matrix along with its inverse is presented for canonical to diagonal transformation. This general result apart from simplifying the process of transformation of linear dynamic systems helps us to reparameterise the models to a desired form.

Determining parameters of dynamic systems from noisy measurements

This paper uses the method of least-squares k-spline approximation to generate time derivatives of sampled- data measurements required to determine values for parameters in models of linear and nonlinear dynamic systems. When identifying the parameter values in a nonlinear system in which signals are corrupted with measurement noise, the method developed in this paper provides more accurate values than does an approximate time-delay filter. In generating the required deriva tives in the presence of noise, the least-squares k- spline approximation yielded higher-order derivatives which were substantially more accurate than the same derivatives obtained from a low-pass filter. In a nonlinear case used as an illustrative example, the technique described in this paper resulted in as much as a 32% improvement in the accuracy of one of the identified model parameter values, and its largest error in any of the six parameters identified was 3.5 percent. The method was particularly effective in counteracting impulse noise caused by bad data points.