Linear Dynamic System Model Research Articles

An approach to logical-mathematical computer modeling of linear and nonlinear dynamical systems

The paper deals with the problem of modeling linear and nonlinear dynamical systems using information technologies and programming. The processes of modeling, synthesis and inference of solutions to problems are considered, and the possibility of their integration within the described approach is shown. The substantive aspects of the development of special mathematical and algorithmic support for the construction of a software system for the analysis and processing of expert information for the purposes of automated synthesis and inference of solutions are briefly outlined. The aim of the work is to create a new approach of logical-mathematical computer modeling of linear and nonlinear dynamical systems and to develop a prototype of a software computational expert system based on it. As a result of the research, a universal approach of logical-mathematical computer modeling of linear and nonlinear dynamical systems has been developed and proposed. The proposed approach can be used to organize the process of computer modeling and automated synthesis of task solutions with the output of their results. The mathematical model of the knowledge base and process of its construction for the selected subject area, as well as the algorithm for outputting solutions to problems with its software implementation in the Python programming language, are shown. Examples of solution output for various linear and nonlinear problems are given. The developed prototype of the software computational expert system using the proposed logical- mathematical approach will significantly simplify the process of modeling dynamical systems and make the solution of problems' simpler and more understandable.

QUASI-ANALYTIC METHOD OF LINEAR DYNAMIC SYSTEMS INVERSION

The problem of inversion of dynamic systems has become widespread while solving the problems of control, identification, and measurement problems arising during the design and research of electrical and mechanical dynamic systems. Inverting is an effective way of implementing disturbance control processes, as well as in combined control systems with a predictive model. The analysis of information sources showed that in the practical solution of most inversion problems, a number of difficulties arise, which are associated with the high sensitivity of the results in relation to the accuracy of the parameters of the mathematical model of the control object, the instability of the inverse model of non-minimum-phase objects, and the violation of the conditions of physical feasibility. The work offers an effective method of inverting linear stationary dynamic systems, free from the mentioned shortcomings in many respects. The basis of the method is the presentation of input and output signals in the form of infinite linear combinations of their derivatives. A method of determining the sequence of matrix coefficients of linear representations of input and output signals is proposed. The main theoretical result is obtaining relationships between matrix coefficients of input and output signals. The work considers mathematical models of linear dynamic systems in the form of differential equations in the state space and in the equivalent "input-output" form. The considered systems must meet the conditions of asymptotic stability, as well as the condition of equal dimensions of the input and output vectors. Requirements for mathematical models of input and output signals are given, the fulfillment of which allows, instead of infinite sums representing signals, to be limited to a finite number of terms.

Probing the Relationship Between Latent Linear Dynamical Systems and Low-Rank Recurrent Neural Network Models

A large body of work has suggested that neural populations exhibit low-dimensional dynamics during behavior. However, there are a variety of different approaches for modeling low-dimensional neural population activity. One approach involves latent linear dynamical system (LDS) models, in which population activity is described by a projection of low-dimensional latent variables with linear dynamics. A second approach involves low-rank recurrent neural networks (RNNs), in which population activity arises directly from a low-dimensional projection of past activity. Although these two modeling approaches have strong similarities, they arise in different contexts and tend to have different domains of application. Here we examine the precise relationship between latent LDS models and linear low-rank RNNs. When can one model class be converted to the other, and vice versa? We show that latent LDS models can only be converted to RNNs in specific limit cases, due to the non-Markovian property of latent LDS models. Conversely, we show that linear RNNs can be mapped onto LDS models, with latent dimensionality at most twice the rank of the RNN. A surprising consequence of our results is that a partially observed RNN is better represented by an LDS model than by an RNN consisting of only observed units.

Virtual sensing for dynamic industrial process based on localized linear dynamical system models with time-delay optimization

Virtual sensors play an important role in real-time sensing of key quality-related variables in industrial processes. Linear dynamical system (LDS) paradigm has established itself as a powerful tool for developing dynamic virtual sensors. However, there are still some practically pivotal issues unresolved, such as how to improve the generalization reliability and accuracy when accounting for the time delays and how to broaden the application sphere by breaking their limitations to linear processes. Motivated by dealing with these challenging issues this paper proposes a virtual sensing framework called ‘localized LDS (LoLDS)’. In the LoLDS framework, the process dynamics and nonlinearities are taken into consideration from different scales without increasing the model complexity, and the time delays are intelligently optimized which triggers the model inconsistency by a designed diversified localization scheme at the offline stage. Moreover, an adaptive online model switch scheme is developed to enable the real-timely best LDS models to be responsible to predict the quality variables. The offline and online operations together enable the LoLDS to improve the generalization performance of the dynamic virtual sensor. The LoLDS framework is highly automated, and its performance has been extensively evaluated by two real-life industrial processes, showing very promising application foregrounds.

Data-Driven Modeling of Linear Dynamical Systems with Quadratic Output in the AAA Framework

We extend the Adaptive Antoulas-Anderson (AAA) algorithm to develop a data-driven modeling framework for linear systems with quadratic output (LQO). Such systems are characterized by two transfer functions: one corresponding to the linear part of the output and another one to the quadratic part. We first establish the joint barycentric representations and the interpolation theory for the two transfer functions of LQO systems. This analysis leads to the proposed AAA-LQO algorithm. We show that by interpolating the transfer function values on a subset of samples together with imposing a least-squares minimization on the rest, we construct reliable data-driven LQO models. Two numerical test cases illustrate the efficiency of the proposed method.

GWO-Based Optimal Tuning of Controllers for Shape Memory Alloy Wire Actuators

The purpose of this paper is to design the optimal controllers for nonlinear processes with Shape Memory Alloy (SMA) wire actuators viewed as controlled processes. Optimal process models are first derived. A comparative analysis is done between the evolved Takagi-Sugeno-Kang (TSK) fuzzy models of SMA wire actuators, two linear dynamic system models with parameters optimally tuned using the recent metaheuristic Grey Wolf Optimiser (GWO) algorithm and one linear dynamic system model obtained in a previous paper using the System Identification Toolbox. Measured input-output data is involved in this comparison. The parameters of two Proportional-Integral-Derivative controllers, a type-1 fuzzy controller and an interval type-2 fuzzy controller are next optimally tuned using GWO, and a comparative analysis of these linear and nonlinear controllers is done. The simulation results illustrate that the type-2 fuzzy controller ensure good performance for these processes.

Block-Wise Parallel Semisupervised Linear Dynamical System for Massive and Inconsecutive Time-Series Data With Application to Soft Sensing

Linear dynamical system (LDS) has established itself as a powerful tool for soft sensing dynamic industrial processes, which however still faces some practically pivotal issues. Firstly, traditional LDS-based soft sensors require training samples to be labeled and ignore time delays (TDs) between explanatory variables (EVs) and primary variables (PVs), whereas labeled samples are usually scarce and the TDs could have appreciable impacts on the predictive accuracy. Secondly, LDS models are trained in a serial way, rendering significant computational deficiency when dealing with massive data that imply a wealth of information on process dynamics. What’s worse, such learning paradigm may fail to take full advantage of all available data, because the time-series data chain is often broken due to malfunctions of data communication system or measurement sensors. These issues lead the LDS-based soft sensors to degraded performance such as low accuracy, computational deficiency and poor interpretability. To this end, this paper first proposes a semisupervised training framework for the LDS with optimized TDs (TD-SsLDS), so as to make up for the deficiencies of supervised learning and overlooking of TDs. Further, a block-wise parallel TD-SsLDS (BwP-TD-SsLDS) is proposed, where a highly efficient learning algorithm is designed to extract patterns from multiple data blocks in a parallel way such that the LDS is enabled to learn from massive and inconsecutive time-series data. Case studies are conducted on both artificial example and real-life industrial process for evaluating the performance of the TD-SsLDS and BwP-TD-SsLDS. The results demonstrate that the proposed schemes could significantly improve the predictive and computational performance for the LDS-based soft sensors.

Понижение порядка сложных моделей с помощью инструментов Robust Control Toolbox

The city of Berezniki, Perm region is located on an underworked mine area. For several years, the city has been experiencing active subsidence of the soil, which provoke the destruction of buil-dings. Therefore, for several years now, the city's buildings and structures have been monitored, which makes it possible to analyze the degree of subsidence. Models of a sufficiently high order are used for an accurate analysis of the situation and forecasting. The article is about a possibility of modeling the deformation of buildings associated with soil subsidence, as a result of mine workings in the city of Berezniki. The purpose of the study is to consider the capabilities of the ‘Robust Control Toolbox’ for reducing the order of complexity of models. An example of an eight-story building included in the collection of reference examples for reducing models of linear dynamic systems is used. Materials and methods. Typical steps for solving the problem of model reduction are presented, commands and tools used to solve this problem are described. The parameters of the model in the state space are determined, which has 48 states, which are displacements or rates of change. The singular values of Hankel are used to select states that can be neglected. The model is reduced using an adaptive error boundary. Reduction using the multiplicative error bound is considered. Comparison of the results of reduction of the model by all described methods is carried out, the choice of the best method of reduction of the model is substantiated. Results. An analysis of the approximation error was performed for all the methods. The maximum relative error has been calculated. An example of calculating the order of the model for a given error value of 5% is given. The order of the result model is 34 states with the error less is then 1%, which is less than the original model. As a result, the AFCs of the original and reduced models, as well as the transient processes of the models, were constructed. The plots in the frequency domain of the models practically coincide, which indicates an adequate description of the system. Conclusions. As a result, it was shown that it is possible to reduce the size of the model by 14 orders of magnitude, goal achieved.

State-space optimal feedback control of optogenetically driven neural activity

Objective. The rapid acceleration of tools for recording neuronal populations and targeted optogenetic manipulation has enabled real-time, feedback control of neuronal circuits in the brain. Continuously-graded control of measured neuronal activity poses a wide range of technical challenges, which we address through a combination of optogenetic stimulation and a state-space optimal control framework implemented in the thalamocortical circuit of the awake mouse. Approach. Closed-loop optogenetic control of neurons was performed in real-time via stimulation of channelrhodopsin-2 expressed in the somatosensory thalamus of the head-fixed mouse. A state-space linear dynamical system model structure was used to approximate the light-to-spiking input-output relationship in both single-neuron as well as multi-neuron scenarios when recording from multielectrode arrays. These models were utilized to design state feedback controller gains by way of linear quadratic optimal control and were also used online for estimation of state feedback, where a parameter-adaptive Kalman filter provided robustness to model-mismatch. Main results. This model-based control scheme proved effective for feedback control of single-neuron firing rate in the thalamus of awake animals. Notably, the graded optical actuation utilized here did not synchronize simultaneously recorded neurons, but heterogeneity across the neuronal population resulted in a varied response to stimulation. Simulated multi-output feedback control provided better control of a heterogeneous population and demonstrated how the approach generalizes beyond single-neuron applications. Significance. To our knowledge, this work represents the first experimental application of state space model-based feedback control for optogenetic stimulation. In combination with linear quadratic optimal control, the approaches laid out and tested here should generalize to future problems involving the control of highly complex neural circuits. More generally, feedback control of neuronal circuits opens the door to adaptively interacting with the dynamics underlying sensory, motor, and cognitive signaling, enabling a deeper understanding of circuit function and ultimately the control of function in the face of injury or disease.

Monitoring and forecasting the COVID-19 epidemic in the UK

This paper shows how existing methods of time series analysis and modeling can be exploited in novel ways to monitor and forecast the COVID-19 epidemic. In the past, epidemics have been monitored by various statistical and model metrics, such as evaluation of the effective reproduction number, R(t). However, R(t) can be difficult and time consuming to compute. This paper suggests two relatively simple data-based metrics that could be used in conjunction with R(t) estimation and provide rapid indicators of how the epidemic’s dynamic behavior is progressing. The new metrics are the epidemic rate of change (RC) and a related state-dependent response rate parameter (RP), recursive estimates of which are obtained from dynamic harmonic and dynamic linear regression (DHR and DLR) algorithms. Their effectiveness is illustrated by the analysis of COVID-19 data in the UK and Italy. The paper also shows how similar methodology, combined with the refined instrumental variable method for estimating hybrid Box–Jenkins models of linear dynamic systems (RIVC), can be used to relate the daily death numbers in the Italian and UK epidemics and then provide 15-day-ahead forecasts of the UK daily death numbers. The same approach can be used to model and forecast the UK epidemic based on the daily number of COVID-19 patients in UK hospitals. Finally, the paper speculates on how the state-dependent parameter (SDP) modeling procedures may provide data-based insight into a nonlinear differential equation model for epidemics such as COVID-19.

A Dynamic Latent Structure With Time-Varying Parameters for Virtual Sensing of Industrial Process With Irregular Missing Data

This paper proposes a novel dynamic latent structure with time-varying parameters for virtual sensing of industrial process with irregular missing data. The proposed latent structure is based on the linear dynamic system (LDS) model. In order to capture the time-varying process characteristics, a Karman filter based parameter updating method is developed and a virtual sensor is constructed to predict hard-to-measure quality variables. The latent variable structure of the improved model enables the virtual sensor to capture the variable cross-correlation and autocorrelation in the missing data by considering both spatial and temporal information, so that the information in the missing data can be learned from those not missing in the samples as well as the Markov process in the temporal stream. Incorporation of both spatial and temporal information renders more flexibility, resulting in a virtual sensor with higher prediction accuracy even with irregular missing data. The better performance of the proposed method is verified by two industrial applications with different ratios of irregular missing data.

Optimizing Signal Timing Control for Large Urban Traffic Networks Using an Adaptive Linear Quadratic Regulator Control Strategy

Traffic signal control is important for intersection safety and efficiency. However, most traffic signal control methods are designed for individual intersections or corridors. Although some adaptive control systems have been developed, the methods used are often proprietary and not published, making it difficult to evaluate their effectiveness. This study proposes an adaptive multi-input and multi-output traffic signal control method that not only can improve network-wide traffic operations in terms of reduced traffic delay and energy consumption, but also is more computationally feasible than existing centralized signal control methods. Considering intersection interactions, a linear dynamic traffic system model was built and adaptively updated to reflect how the signal control input of each intersection affects network-wide vehicle travel delay. Based on the system model, an adaptive linear-quadratic regulator (LQR) was designed to minimize both traffic delay and incremental changes in the control input. The proposed control method was evaluated in a microscopic traffic simulation environment with a 35-intersection network of Bellevue City, Washington. Simulation results show that the proposed method had shorter average traffic delays in the network when compared with the traffic delays controlled by the state-of-the-art max-pressure, self-organizing traffic lights, and independent deep Q network methods.

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.

Dynamic identification of boundary conditions for convection-diffusion transport model subject to noisy measurements

The paper addresses the problem of dynamic identification of boundary conditions for one-dimensional convection-diffusion transport model based on noisy measurements of the function of interest. Using finite difference method the original model with the partial differential equation is replaced with the discrete linear dynamic system model with noisy multisensor measurements in which boundary conditions are included as the unknown input vector. To solve the problem, the algorithm of simultaneous the state and input vector estimation is used. The results of numerical experiments are presented which confirm the practical applicability of the proposed approach.

Practical Tools and Guidelines for Exploring and Fitting Linear and Nonlinear Dynamical Systems Models

A dynamical system is a system of variables that show some regularity in how they evolve over time. Change concepts described in most dynamical systems models are by no means novel to social and behavioral scientists, but most applications of dynamic modeling techniques in these disciplines are grounded on a narrow subset of—typically linear—theories of change. I provide practical guidelines, recommendations, and software code for exploring and fitting dynamical systems models with linear and nonlinear change functions in the context of four illustrative examples. Cautionary notes, challenges, and unresolved issues in utilizing these techniques are discussed.

Chronic SSRI administration in Lsamp deficient mice

This paper presents an algorithm for the localization of forces applied to a structure. The force estimation is performed in the frequency domain based on a limited number of sensors and a linear dynamic system model and it involves minimizing an objective function penalized with a group sparsity term. The minimization of this objective function is formulated as a second order cone program, which is solved using an interior point method. This allows for a reduction of the calculation time when compared to other algorithms that are currently available to enforce group sparsity on the forces, especially for large scale problems. The presented algorithm is first verified using numerical simulations. Next, a validation is performed using data obtained from a field test on a footbridge, where two locations on the bridge deck are excited using hammer impacts and the force localization is performed assuming a total of 108 possible force locations.

Reduced order modeling of random linear dynamical systems based on a new a posteriori error bound

SummaryReduced order models (ROMs) are becoming increasingly useful for saving computational cost in response prediction of vibrating systems. In a number of applications such as uncertainty quantification, ROMs require robustness over a wide variation of parameters. Accordingly, often they are classified as local and global, based on their performance in the parametric domain. Availability of an error bound of a ROM helps in achieving this robustness, mainly by allowing adaptivity. In this work, for a linear random dynamical system, first, an a posteriori error bound is developed based on the residual in the governing differential equation. Next, based on this error bound, two adaptive methods are proposed for building robust ROMs, that is, one for local, and another for global. While both methods are based on a greedy search approach, a modification is proposed in the training stage of the global ROM for accelerated convergence. These methods are then applied to an uncertainty quantification problem in a statistical simulation framework, and accordingly, two algorithms are developed. A detailed numerical study on vibration of a bladed disk assembly is conducted to study the accuracy and efficiency of the proposed error bound and adaptive ROMs. It is found that these adaptive ROMs provide a considerable speed‐up in estimating the probability of failure.

Localization of dynamic forces on structures with an interior point method using group sparsity

Abstract This paper presents an algorithm for the localization of forces applied to a structure. The force estimation is performed in the frequency domain based on a limited number of sensors and a linear dynamic system model and it involves minimizing an objective function penalized with a group sparsity term. The minimization of this objective function is formulated as a second order cone program, which is solved using an interior point method. This allows for a reduction of the calculation time when compared to other algorithms that are currently available to enforce group sparsity on the forces, especially for large scale problems. The presented algorithm is first verified using numerical simulations. Next, a validation is performed using data obtained from a field test on a footbridge, where two locations on the bridge deck are excited using hammer impacts and the force localization is performed assuming a total of 108 possible force locations.

Sparse Bayesian ARX models with flexible noise distributions

This paper considers the problem of estimating linear dynamic system models when the observations are corrupted by random disturbances with nonstandard distributions. The paper is particularly motivated by applications where sensor imperfections involve significant contribution of outliers or wrap-around issues resulting in multi-modal distributions such as commonly encountered in robotics applications. As will be illustrated, these nonstandard measurement errors can dramatically compromise the effectiveness of standard estimation methods, while a computational Bayesian approach developed here is demonstrated to be equally effective as standard methods in standard measurement noise scenarios, but dramatically more effective in nonstandard measurement noise distribution scenarios.

Tensor-based linear dynamical systems for action recognition from 3D skeletons

Recent years have seen a growth in interest in skeleton-based human behavior recognition. Skeleton sequences can be expressed naturally as high-order tensor time series, and in this paper we report on the modeling and analysis of such time series using a linear dynamical system (LDS). Owing to their relative simplicity and efficiency, LDSs are the most common tool used in various disciplines for encoding spatiotemporal time series data. However, conventional LDSs process the latent and observed states at each frame of a video as a column vector, a representation that fails to take into account valuable structural information associated with human action. To correct this, we propose a tensor-based linear dynamical system (tLDS) for modeling tensor observations in time series and employ Tucker decomposition to estimate the parameters of the LDS model as action descriptors. In this manner, an action can be expressed as a subspace corresponding to a point on a Grassmann manifold on which classification can be performed using dictionary learning and sparse coding. Experiments using the MSR Action3D, UCF Kinect, and Northwestern-UCLA Multiview Action3D datasets demonstrate the excellent performance of our proposed method.