Linear State-space Systems Research Articles

Analyzing asymmetry in brain hierarchies with a linear state-space model of resting-state fMRI data.

This study challenges the traditional focus on zero-lag statistics in resting-state functional magnetic resonance imaging (rsfMRI) research. Instead, it advocates for considering time-lag interactions to unveil the directionality and asymmetries of the brain hierarchy. Effective connectivity (EC), the state matrix in dynamical causal modeling (DCM), is a commonly used metric for studying dynamical properties and causal interactions within a linear state-space system description. Here, we focused on how time-lag statistics are incorporated within the framework of DCM resulting in an asymmetric EC matrix. Our approach involves decomposing the EC matrix, revealing a steady-state differential cross-covariance matrix that is responsible for modeling information flow and introducing time-irreversibility. Specifically, the system's dynamics, influenced by the off-diagonal part of the differential covariance, exhibit a curl steady-state flow component that breaks detailed balance and diverges the dynamics from equilibrium. Our empirical findings indicate that the EC matrix's outgoing strengths correlate with the flow described by the differential cross covariance, while incoming strengths are primarily driven by zero-lag covariance, emphasizing conditional independence over directionality.

On BIBO Stability of Infinite-Dimensional Linear State-Space Systems

On BIBO Stability of Infinite-Dimensional Linear State-Space Systems

Mass and Force Lumping: An Essential Enhancement to the Intrinsic Beam Finite Element Discretization

This paper introduces the novel application of the mass and force lumping technique to enhance the finite element discretization of the fully intrinsic beam formulation. In our aeroelastic system model, 2-D unsteady aerodynamics were incorporated alongside simple calculations for thrust and gravity. Through the central difference discretization method, the discretized system was thoroughly examined, shedding light on the advantages of the mass and force lumping approach. With the use of a first-order lumping method, we successfully reconstructed the inertia matrices, external forces, and moments. The resulting equations are more systematically structured, facilitating the extraction of a regular state-space linear system using the direct index reduction method post-linearization. Numerical results further confirm that the proposed techniques can effectively capture the nonlinear dynamics of aeroelastic systems, enabling equation reconstruction and leading to significant benefits in system order reduction and flight dynamical analysis.

Using graphics processing units for the efficient dynamic simulation of wind farms

Dynamic simulations are crucial for proper power system operation. However, these can be computationally expensive. This article introduces an efficient CUDA kernel for GPU acceleration, that achieves parallelization without the need to solve many different power systems at once. The proposal unifies generator and grid execution within a single CUDA kernel, ensuring accurate dynamic simulations of wind farms. A custom solution from scratch was developed, that integrates the power system network solver with connected devices. Wind turbines are modeled as state-space linear systems, while the grid is parallelized through subunit partitioning and small submatrix inverses to reduce computational costs. The resulting kernel extracts parallelism in grids with limited inherent parallelism, outperforming previous methods such as column-level parallelism. Simulations on an NVIDIA GeForce RTX 3090 show a 26.39x speedup with respect to the C++ version, and the cooperative groups' synchronization primitives provide optimal performance while simulating up to 2464 wind turbines.

Global Temperature Projections from a Statistical Energy Balance Model Using Multiple Sources of Historical Data

Abstract This paper estimates a two-component energy balance model as a linear state-space system (EBM-SS model) using historical data. It is a joint model for the temperature in the mixed layer, the temperature in the deep ocean layer, and radiative forcing. The EBM-SS model allows for the modeling of nonstationarity in forcing and the incorporation of multiple data sources for the unobserved processes. We estimate the EBM-SS model using historical datasets at the global level for the period 1955–2020 by maximum likelihood. We show in the empirical estimation and in simulations that using multiple data sources for the unobserved processes reduces parameter estimation uncertainty. When fitting the EBM-SS model to six observational global mean surface temperature (GMST) anomaly series, the GMST projections under representative concentration pathway scenarios are comparable to those from Coupled Model Intercomparison Project models. The results show that a simple statistical climate model estimated on the historical period can produce GMST projections compatible with output from large-scale Earth system models. Significance Statement We develop a statistical model to understand Earth’s energy balance and how it impacts temperature across the planet’s surface and deep oceans. Unlike previous approaches, ours is fully statistical. This means that we use maximum likelihood to calculate the estimates and evaluate the level of uncertainty in the model’s parameters. By applying the model to historical data from 1955 to 2020, we are able to make predictions about the average global surface temperature that are consistent with those from the Coupled Model Intercomparison Project. Our findings support current predictions about global temperatures using a straightforward statistical climate model, grounded in historical data.

Revisiting State Estimation for Linear Systems with State and/or Filter Equality Constraints*

This article revisits optimal state estimation, in the mean squared error matrix sense, for linear systems with state and/or filter subject to linear equality constraints (LECs). First, it is shown that the conventional Wiener filter (WF) form incorporates any LECs on the state, thus yielding a filter subject to the same LECs. Conversely, an optimal linear filter subject to LECs (or linear equality gain constraints) in general does not exist. Therefore, adding LECs on the WF or WF gain matrix either leaves unchanged or degrades the constrained WF performance w.r.t. the unconstrained WF. Since the Kalman filter (KF) and Kalman predictor (KP) are recursive WF forms for linear discrete state-space (LDSS) systems, the same results hold for both estimators, which is in contradiction with several existing results in the literature. Actually, even if these existing results are mathematically correct, however, they have been derived for unsuitable assumed LDSS models where the state is surprisingly not compliant with the assumed LECs. Indeed, it is shown that for suitable assumed state models, both standard KF and KP forms satisfy the assumed LECs, making any additional projection step superfluous.

Monte Carlo investigation of structural minimality for structures of uncontrolled linear switching systems with Maple

One path to understanding a physical system is to represent it by a model structure (collection of related models). Suppose our system is not subject to external influences, and depends on unobservable state variables (x), and observables (y). Then, a suitable uncontrolled, state-space model structure S is defined by relationships between x and y, involving parameters θ ∈ Θ. That is, each parameter vector in parameter space Θ is associated with a particular model in S.
 Before using S for prediction, we require system observations for parameter estimation. This process aims to determine θ values for which predictions “best” approximate the data (according to some objective function). The result is some number of estimates of the true parameter vector, θ*. Multiple parameter estimates are problematic when these cause S to produce dissimilar predictions beyond our data's range. This can render us unable to confidently make predictions, resulting in an uninformative study.
 Non-uniqueness of parameter estimates follows when S lacks the property of structural global identifiability (SGI). Fortunately, we may test S for SGI prior to data collection. The absence of SGI encourages us to rethink our experimental design or model structure.
 Before testing S for SGI we should check that it is structurally minimal. If so, we cannot replace S by a structure of fewer state variables which produces the same output. Most testing methodology is applicable to structures which employ the same equations for all time. These methods are not appropriate when, for example, a process has an abrupt change in its dynamics. For such a situation, a structure of linear switching systems (LSSs) may be suitable. Any system in the structure has a collection of linear time-invariant state-space systems, and a switching function which determines the system in effect at each instant. As such, we face a novel challenge in testing an LSS structure for SGI.
 We will consider the case of an uncontrolled LSS structure of one switching event (a ULSS-1 structure). In this setting, we may approach the structural minimality problem via the Laplace transform of the output function on each time interval. Each rational function yields conditions for pole-zero cancellation. If these conditions are not satisfied for almost all θ ∈ Θ, then S is structurally minimal.
 Analytical approaches can be quite laborious. However, we may expect a numerical approach to provide useful insights quickly. For example, if pole-zero cancellation occurs for almost all of a sufficiently large number of parameter values sampled from Θ, then structural minimality is possible. This result may encourage us to prove the existence of structural minimality.
 We shall use Maple 2020-2 to conduct a numerical investigation of structural minimality for a test case ULSS-1 structure applicable to flow-cell biosensor experiments used to monitor biochemical interactions, which include the popular Biacore-branded units.

Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control

The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

Numerical Solution for Linear State Space Systems using Haar Wavelets Method

In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution.

An Integro-differential Operator Approach to Linear State-space Systems

In this paper, the algebraic analysis approach to linear state-space systems is further developed using rings of integro-differential operators. The module structure of linear state-space systems is investigated over these rings. The module associated with a linear state-space system is shown to be the direct sum of the stably free module defined by the linear system without inputs and the free module defined by the inputs of the system.

Bandwidth Expansion of Atomic Spin Gyroscope With Transient Response

Atomic spin gyroscopes have ultra-high sensitivity on inertial measurement, but the narrow bandwidth limits its application. We describe a bandwidth expansion method for atomic spin gyroscope based on the transient response of electron spin. The gyroscope is modeled as a four inputs two outputs linear state-space system by commonly used linearization. As the transverse component of electron spin polarization decays much faster than that of nuclear spin, a quasi-steady-state approximation on electron spin is proposed to estimate the transverse polarization component of nuclear spin in real-time. In experiment, the system matrix, input matrix, and output matrix of the state-space model are decided with a series of calibrations. A bandwidth expansion method is proposed to solve the rotation input with the transient response of the electron spin. The proposed method shows strong rotation tracking ability compared with the steady-state method. The bandwidth of atomic spin gyroscope is extended and 3 dB/Oct lower signal attenuation in the frequency range of 0.4~5 Hz is obtained with the proposed method.

An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

In this paper, we further develop the study of rings of integro-differential-delay operators considered as noncommutative polynomial algebras satisfying standard calculus identities. Within the algebraic analysis approach, we show that transformations and reductions of linear differential time-delay systems can be interpreted as homomorphisms and isomorphisms of finitely presented left modules over an algebra of integro-differential-delay operators. In particular, we show how Fiagbedzi-Pearson's transformation can be found again and generalized. This transformation maps the solutions of a first-order differential linear system with state and input delays to the solutions of a purely state-space linear system. Fiagbedzi-Pearson's transformation reduces to the well-known Artstein's reduction when the system has no state delay and yields an isomorphism of the solution spaces.

Dynamic Expectation Maximization Algorithm for Estimation of Linear Systems with Colored Noise

The free energy principle from neuroscience has recently gained traction as one of the most prominent brain theories that can emulate the brain’s perception and action in a bio-inspired manner. This renders the theory with the potential to hold the key for general artificial intelligence. Leveraging this potential, this paper aims to bridge the gap between neuroscience and robotics by reformulating an FEP-based inference scheme—Dynamic Expectation Maximization—into an algorithm that can perform simultaneous state, input, parameter, and noise hyperparameter estimation of any stable linear state space system subjected to colored noises. The resulting estimator was proved to be of the form of an augmented coupled linear estimator. Using this mathematical formulation, we proved that the estimation steps have theoretical guarantees of convergence. The algorithm was rigorously tested in simulation on a wide variety of linear systems with colored noises. The paper concludes by demonstrating the superior performance of DEM for parameter estimation under colored noise in simulation, when compared to the state-of-the-art estimators like Sub Space method, Prediction Error Minimization (PEM), and Expectation Maximization (EM) algorithm. These results contribute to the applicability of DEM as a robust learning algorithm for safe robotic applications.

On σ-entropy Analysis of Linear Stochastic Systems in State Space

In this paper the problem of random disturbance attenuation capabilities in linear continuous systems is studied. It is supposed that the system operates under random disturbances with bounded σ-entropy level. σ-entropy norm indicates a performance index of the continuous system on the set of the random signals with bounded σ-entropy. This paper presents a time-domain solution to the calculation of σ-entropy norm of the continuous linear time-invariant system. σ-entropy norm is defined after solving coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations.

Control of inventory system with random demand and product damage during delivery using the linear quadratic gaussian method

This study formulates a dynamical system for the control of a single product inventory system in accordance with the random value of demand and the percentage of damaged product during the delivery process. The formulated model has the form of a linear state-space system comprising of two disturbances, which represents the random value of demand and the percentage of the damaged product during delivery. The optimal value of the product amount ordered to the supplier is properly calculated by using the linear quadratic gaussian (LQG) method. The controller is used by the manager to make inventory level decisions under the uncertainty of demand and damaged items during the product delivery process. The result showed that the optimal product order for each review time was achieved, and the inventory level was used to obtain the right set point properly. Moreover, based on comparison with other research results, the proposed model was well performed.

An improved single-particle model with electrolyte dynamics for high current applications of lithium-ion cells

In this work, an isothermal, reduced-order polynomial-based model is proposed for lithium-ion cells. The correction incorporated into the single-particle model considers the effect of electrolyte dynamics leading to improved model predictions at high current applications without adding any significant computational complexity. The model includes the spatial distribution of the electrode and electrolyte potentials and the electrolyte lithium concentration on the resulting cell voltage. A novel approach to account for the spatially varying overpotential and open-circuit potential is proposed. A linear system of differential-algebraic equations is obtained in the state variables and the cell voltage is computed using a non-linear expression in these states. The resulting linear state-space system makes the model easily implementable for state-estimation algorithms in battery management system applications. Additionally, a semi-analytical model is incorporated for the diffusion of lithium in electrodes and a criterion for truncating the infinite-series solution is proposed for achieving a balance between accuracy and complexity for cells subjected to fluctuating currents. The proposed model results in a decrease in error for cell voltage prediction by a factor of five when compared with existing enhanced single-particle models.

On Bayesian active vibration control of structures subjected to moving inertial loads

This study introduces a novel Bayesian framework for online and real-time vibration control of beam type structures, which represent a comprehensive control system associated with input-state algorithms. Control design systems typically require knowledge of system states, which in structures are displacements and velocities at some degrees of freedom. Currently, full-field measurements of displacements and velocities in large structural systems are not feasible. Also, properties of the moving inertial loads as key parameters in control designs are assumed known; however, in practice, measuring their characteristics is a challenging issue. As a remedy, an observer is required to estimate quantities of interest needed in control algorithms via indirect measurement of structural response. To estimate the entire state of the system in the absence of input information, often the input needs to be estimated together with the unknown states. Three state of the art methods for estimation of input and states of a linear state-space system, namely augmented Kalman filter (AKF), dual Kalman filter (DKF), and joint input state estimation algorithm (JIS), were adopted to assess controller performance in the absence of direct displacement measurements in comparison to conventional Kalman filter (KF). The linear quadratic Gaussian (LQG) framework, in association with these three filters, presents a novel control algorithm. Various sensor networks featuring pure accelerations and a combination of strain gauges and accelerometers with different noise levels were considered. Extensive numerical examples and comparison with the passive control system proved the viability of input-state estimation as an integral part of active control algorithms in structural systems subjected to moving inertial loads with different speeds.

Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions

<p style='text-indent:20px;'>The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.</p>

An Outlier Robust Finite Impulse Response Filter With Maximum Correntropy

Non-Gaussian noise is common in industrial applications, and it is a severe challenge to existing state estimators. In this paper, a novel robust maximum correntropy finite impulse response (MCFIR) filter is proposed to deal with the state estimation problem in the linear state-space system corrupted by outliers. The filter operates as a finite memory form, and thus it obtains superior immunity to noise statistics and process uncertainties than existing Kalman-like robust filters. Gaussian correntropy is adopted to generate a new cost function, which improves the filter robustness to outlier interference. We derive an unbiased MCFIR filter that ignores noise statistics and propose an improvement bias-constrained MCFIR filter to achieve better estimate accuracy. To improve the filtering performance degradation caused by improper kernel size, an adaptive kernel size algorithm is further proposed, which adjusts the bandwidth within a specific range adaptively and achieves significant improvement in the MCFIR filter. An illustrative example based on moving target tracking is presented to evaluate the performance of the proposed filter, and simulation results confirmed that the MCFIR filter obtained superior immunity to outliers than the existing robust filters.

Fluid temperature predictions of geothermal borefields using load estimations via state observers

ABSTRACT Fluid temperature predictions of geothermal borefields usually involve temporal superposition of its characteristic g-function, using load aggregation schemes to reduce computational times. Assuming that the ground has linear properties, it can be modelled as a linear state-space system where the states are the aggregated loads. However, the application and accuracy of these models is compromised when the borefield is already operating and its load history is not registered or there are gaps in the data. This paper assesses the performance of state observers to estimate the borefield load history to obtain accurate fluid predictions. Results show that both Time-Varying Kalman Filter (TVKF) and Moving Horizon Estimator (MHE) provide predictions with average and maximum errors below 0.1C and 1C, respectively. MHE outperforms TVKF in terms of n-step ahead output predictions and load history profile estimates at the expense of about five times more computational time.