State Space Of System Research Articles

BLESS Behavior Correctness Proof as ConvincingVerification Artifact

Safety-critical cyber-physical systems require evidence they are indeed safe. In practice, such evidence is results of system tests. Unfortunately, tests can only demonstrate the presence of software errors, not their absence, and can practically cover a tiny fraction of system state space. Valiant efforts to formally verify program correctness have either been excruciatingly difficult (theorem proving) or incomplete (static analysis, model checking, SAT or SMT solving). The BLESS Methodology was created specifically for engineers in industry to formally verify software controlling machines. The BLESS Methodology transforms programs that control machines, annotated with assertions to form proof outlines, into deductive proofs that every possible program execution will conform to its specification. To the extent that cyber-physical system specifications have been validated to express system safety and performance, a deductive proof can be a convincing argument to a person of program correctness. This paper uses a simple safety-critical system to argue that behavior correctness proof under the BLESS Methodology is a convincing verification artifact in addition to customary testing.

Transportation characteristics analysis of the dense-phase gas-solid two-phase flow based on phase space reconstruction

The operational efficiency and safety of the dense phase pneumatic conveying system is deeply influenced by the transportation characteristics of the dense-phase gas-solid two-phase flow in conveying pipelines. This paper presents a framework for analyzing the transport characteristics of the gas-solid two-phase flow based on the phase space reconstruction. The framework explores the transport characteristics by reconstructing the state evolution trajectories of the two-phase flow system in state space. The main goal is to enhance the interpretability of feature analysis. By employing the phase space reconstruction method to process the output signals of an electrostatic sensor, the evolution trajectory of the dynamic state of the gas-solid two-phase flow system was reconstructed in phase space.This provides an effective entry point for extracting the essential dynamic characteristics of the gas-solid two-phase flow system. Furthermore, a series of feature values were proposed, including the average speed of the trajectory, Shannon entropy of the trajectory, and the average distance between state points in phase space, which were used to respectively evaluate important transportation characteristics such as flowability, stability, and particle aggregation degree of the gas-solid two-phase flow. Moreover, through comparative analysis with visualization images that represent actual conveying conditions, the effectiveness of these proposed feature values in evaluating these transportation characteristics was verified.

Expected First Occurrence Time of Uncertain Future Events in One-Dimensional Linear Systems

The rapid advancement of machine learning algorithms has significantly enhanced tools for monitoring system health, making data-driven approaches predominant in Prognostics and Health Management (PHM). In contrast, model-based approaches have seen modest progress, as they are often constrained by the need for prior knowledge of specific governing equations, limiting their applicability to a wide range of problems. Recently, rigorous theoretical foundations have been established to extend dynamical systems theory by incorporating prognosis of uncertain events. This article leverages this formal framework to introduce and demonstrate a fundamental mathematical result for one-dimensional linear dynamical systems. The presented theorem offers an exact expression for calculating the expected time at which an event will first occur in the future. Unlike typical thresholds, this event is triggered by a hazard zone, defined as an uncertain event likelihood function over the system's state space. Applications of this theorem can be found in implementing real-time prognostic frameworks, where it is crucial to quickly estimate the magnitude of impending failures. Emphasis is placed on minimizing computational burden to facilitate prognostic decision-making.

Nonessentiality of Reservoir's Fading Memory for Universality of Reservoir Computing.

This article describes a novel sufficient condition concerning approximations with reservoir computing (RC). Recently, RC using a physical system as the reservoir has attracted attention. Because many physical systems are modeled as state-space systems, it is necessary to guarantee the approximations given by reservoirs represented as nonlinear state-space systems. There are two problems with existing approaches: a reservoir must have a property called fading memory and must be represented as a set of maps between input and output signals on the bi-infinite-time (BIT) interval. These two conditions are too strict for reservoirs represented as nonlinear state-space systems as they require the reservoir to have a unique equilibrium state for the zero input. This article proposes an approach that employs operators from right-infinite-time (RIT) inputs to RIT outputs. Furthermore, we develop a novel extension of the Stone-Weierstrass theorem to handle discontinuous functions. To apply the extended theorem, we define functionals corresponding to operators and introduce a metric on the domain of the functionals. The resulting sufficient condition does not require the reservoir to have fading memory or continuity with respect to inputs and time. Therefore, our result guarantees the approximations with very common reservoirs and provides a rationale for physical RC. We present an example of a physical reservoir without fading memory. With the example reservoir, the RC model successfully approximates NARMA10, a benchmark task for time series predictions.

A mutual information statistic for assessing state space partitions of dynamical systems.

We propose a mutual information statistic to quantify the information encoded by a partition of the state space of a dynamical system. We measure the mutual information between each point's symbolic trajectory history under a coarse partition (one with few unique symbols) and its partition assignment under a fine partition (one with many unique symbols). When applied to a set of test cases, this statistic demonstrates predictable and consistent behavior. Empirical results and the statistic's formulation suggest that partitions based on trajectory history, such as the ordinal partition, perform best. As an application, we introduce the weighted ordinal partition, an extension of the popular ordinal partition with parameters that can be optimized using the mutual information statistic, and demonstrate improvements over the ordinal partition in time series analysis. We also demonstrate the weighted ordinal partition's applicability to real experimental datasets.

Workpiece temperature control in friction stir welding of Inconel 718 through integrated numerical analysis and process control

Friction stir welding (FSW) offers significant advantages over fusion welding, particularly for high-strength alloys like Inconel 718. However, achieving optimal surface quality in Inconel 718 FSW remains challenging due to its sensitivity to temperature fluctuations during welding. This study integrates finite element simulations, statistical analysis, and advanced control methodologies to enhance weld surface quality through adequate thermal management. High-fidelity simulations of the FSW process were conducted using a validated 3D transient COMSOL Multiphysics model, producing a comprehensive dataset correlating process parameters (rotational speed, axial force, and welding speed) with workpiece temperature. This dataset facilitated statistical analysis and parameter optimization through Analysis of variance (ANOVA) method, leading to a deeper understanding of process variables. A nonlinear state-space system model was subsequently developed using experimental data and the system identification toolbox in Matlab, incorporating domain-specific insights. This model was rigorously validated with an independent dataset to ensure predictive accuracy. Utilizing the validated model, tailored control strategies, including proportional-integral-derivative (PID) and model predictive control (MPC) in both single and multivariable configurations, were designed and evaluated. These control strategies excelled in maintaining welding temperatures within optimal ranges, demonstrating robustness in response times and disturbance handling. This precision in thermal management is poised to significantly refine the FSW process, enhancing both surface integrity and microstructural uniformity. The strategic implementation of these controls is anticipated to substantially improve the quality and consistency of welding outcomes.

Variable frequency interharmonic domain methodology for transient analysis and simulation of distributed generation systems

As the integration of renewable energy resources into the power grid increases, accurate modeling of variable frequency distributed generation systems becomes essential for ensuring grid stability, reliability, and optimal operation. Existing models often fail in effectively capturing the frequency dynamics of these systems, particularly when dealing with variable frequency sources. This gap hinders comprehensive analysis and the development of effective planning, control, and mitigation strategies.To address this challenge, this paper introduces a novel modeling methodology named here as the variable frequency interharmonic domain (VFID), which is based on the flexible extended harmonic domain (FEHD) approach. VFID dynamically adjusts the set of pre-selected frequencies within the state-space system to reflect changes in variable frequency sources. This method eliminates the need for post-processing routines, accurately representing both instantaneous values and frequency evolution of time-varying harmonics and interharmonics.Key results demonstrate that VFID offers significant improvements in simulation accuracy over conventional FEHD and PSCAD methods, making it a valuable tool for analyzing and simulating modern distributed generation systems. This advancement not only fills a critical research gap but also contributes to the development of more resilient and sustainable energy infrastructures.

A model of task-level human stepping regulation yields semistable walking.

A simple lateral dynamic walker, with swing leg dynamics and three adjustable input parameters, is used to study how motor regulation affects frontal-plane stepping. Motivated by experimental observations and phenomenological models, we imposed task-level multi-objective regulation targeting the walker's optimal lateral foot placement at each step. The regulator prioritizes achieving step width and lateral body position goals to varying degrees by choosing a mixture parameter. Our model thus integrates a lateral mechanical template, which captures the fundamental mechanics of frontal-plane walking, with a lateral motor regulation template, an empirically verified model of how humans manipulate lateral foot placements in a goal-directed manner. The model captures experimentally observed stepping fluctuation statistics and demonstrates how linear empirical models of stepping dynamics can emerge from first-principles nonlinear mechanics. We find that task-level regulation gives rise to a goal-equivalent manifold in the system's extended state space of mechanical states and inputs, a subset of which contains a continuum of period-1 gaits forming a semistable set: perturbations off of any of its gaits result in transients that return to the set, though typically to different gaits.

Robust PID control of a two-rotor aerodynamical system with decoupling

Abstract In this work, robust PID design via μ synthesis for controlling a MIMO two-rotor aerodynamic system manufactured by the Polish company Inteco is presented. The controller consists of a decoupling system and two PID controllers. The aim is to demonstrate an approach enabling further fine-tuning based on an already synthesized controller structure. Tools from Robust Control Toolbox in MATLAB are used. The PIDs are constructed as tunablePID objects, and the decoupler is also tunable and has the preset structure of a static gain matrix (tunableGain) or dynamic state-space system with a given number of states (tunableSS). The model contains some parameters with structured uncertainty. Several variants of the controller with a fixed structure have been created, and their performance has been tested by simulation and on the laboratory set-up.

System identification for structural condition assessment: Application to critical neoclassical monuments in Nepal

This paper presents structural condition assessment of some critical neoclassical monuments in Nepal using nonparametric and parametric system identification techniques. Neoclassical buildings in Nepal house critical administrative units. Structural condition assessment of such buildings is therefore important for the safety of their occupants and interrupted function during and after major earthquakes. The historical and cultural values of such monuments necessitate regular structural assessment for maintenance, repair, and preservation. Dynamic characteristics such as natural vibration frequencies of structures provide valuable insights on their structural condition. While numerical models are commonly used to estimate eigen frequencies of structures, they are associated with large uncertainties in massive monuments with complex structural and geometrical configurations. Noninvasive dynamic identification techniques provide an alternative means in such structures. Ambient vibration records from six neoclassical monuments in the Kathmandu Valley, Nepal are used in this study to estimate vibration frequencies of the structures in different states. The structures were damaged during the 2015 Gorkha earthquake, and subsequently retrofitted. Changes in vibration frequencies before and after retrofitting provide useful insights on structural improvement through retrofitting. Input-output and output-only system identification techniques are tested to simplify the structural condition assessment approach. We conclude that the state space system identification can stably quantify dynamic properties so stiffness variation can be confidently extracted, which is the key application for noninvasive structural system identification. Also, using minimum number of sensors, we captured damage aggravation deploying the modal assurance criterion. The outcomes indicate that structural condition assessment of complex structures is possible using a limited number of sensors for circumstances such as damage aggravation to temporal variation (evolution/reduction) of dynamic characteristics.

Learnability of state spaces of physical systems is undecidable

Despite an increasing role of machine learning in science, there is a lack of results on limits of empirical exploration aided by machine learning. In this paper, we construct one such limit by proving undecidability of learnability of state spaces of physical systems. We characterize state spaces as binary hypothesis classes of the computable Probably Approximately Correct learning framework. This leads to identifying the first limit for learnability of state spaces in the agnostic setting. Further, using the fact that finiteness of the combinatorial dimension of hypothesis classes is undecidable, we derive undecidability for learnability of state spaces as well. Throughout the paper, we try to connect our formal results with modern neural networks. This allows us to bring the limits close to the current practice and make a first step in connecting scientific exploration aided by machine learning with results from learning theory.

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.

Multiple-model state-space system identification with time delay using the EM algorithm

For a dynamic process identification throughout the whole operating range under diverse operating conditions, it is difficult to capture the process dynamics by a single process model in which the traditional identification method can be adopted to implement parameter estimation. By using the multiple dual-rate state-space models to approach the parameter-varying time-delay systems with different operating conditions, this paper explores an EM algorithm to simultaneously estimate the hidden variable, the parameter vector, the state variable and the time-delay by introducing hidden variables and by using a Kalman smoother.

A model of task-level human stepping regulation yields semistable walking.

A simple lateral dynamic walker, with swing leg dynamics and three adjustable input parameters, is used to study how motor regulation affects frontal plane stepping. Motivated by experimental observations and phenomenological models, we imposed task-level multiobjective regulation targeting the walker's optimal lateral foot placement at each step. The regulator prioritizes achieving step width and lateral body position goals to varying degrees by choosing a mixture parameter. Our model thus integrates a lateral mechanical template, which captures fundamental mechanics of frontal-plane walking, with a lateral motor regulation template, an empirically verified model of how humans manipulate lateral foot placements in a goal-directed manner. The model captures experimentally observed stepping fluctuation statistics and demonstrates how linear empirical models of stepping dynamics can emerge from first-principles nonlinear mechanics. We find that task-level regulation gives rise to a goal equivalent manifold in the system's extended state space of mechanical states and inputs, a subset of which contains a continuum of period-1 gaits forming a semistable set: perturbations off of any of its gaits result in transients that return to the set, though typically to different gaits.

System identification based on the cross Gramian

In order to make mechanical structures more reliable, research is being carried out into methods of detecting and localizing damage to structures, known as Structural Health Monitoring (SHM). One method for continuously monitoring a structure is called State Projection Estimation Error (SP2E) and was developed at the University of Applied Sciences Leipzig. This method is based on state space systems, which are generated by output-only system identification. It is state-of-the-art to create a stabilization diagram to choose the best model order. Since the monitoring process is continuous, a state space model is created every 10 minutes, and therefore it is not possible to look at a stabilization diagram. Thus, the system identification process has to be automated. A novel approach developed in the SPP 100+ research program takes place via the theory of the cross Gramian. The cross Gramian is defined by the Hankel operator, which maps the past input to the future output. This Gramian indicates the interaction between the states and the input/output of a system and can be seen as the energy capacity of a system. By comparing the power of the measurement with the identified energy capacity, the model order selection can be automated. There are limits to the existence of the cross Gramian for multiple-input, multiple-output (MIMO) systems. The system has to be square, stable and symmetric. The covariance driven system identification never provides a symmetric system. A way to create a symmetric system out of the asymmetric measurement data is presented by the use of a symmetrizer, such that the cross Gramian can always be computed. With this symmetrizer a method to estimate the system matrix A by a stable Lyapunov equation, which is classically computed by a numerically unstable pseudo inverse, is presented.

Experimental vibration-based output-only damage localization of mechanical systems

Vibration-based Structural Health Monitoring (SHM) is becoming increasingly widespread in a variety of engineering sectors as its variability enables diverse applications. It enables full-time assessment of the status of a structure, allowing for early maintenance and repair to prevent further damage or even failure. The presented study is part of the SPP100+ program, which has the objective of extending the usability of complex structures. This study presents an experimental investigation of the damage localization of mechanical structures while considering varying environmental and operational conditions (EOC). For this, a testing field was designed and applied, enabling one to study the effects of the varying conditions in a comparable way. The research is based on the approach to modify the state projection estimation error (SP2E) method to account for the influence of EOC. The proposed method utilizes output-only system identification and state space parameterization to estimate the current structural state. Stochastic subspace identification (SSI) and H_∞ -estimation are employed to identify the state space model during a learning phase. Subsequently, the SP2E method is applied during the monitoring phase to localize structural alterations. Moreover, an algorithm is introduced to estimate the weighting factors of different environmental and operational conditions. To verify the efficacy of the proposed method, an extensive experimental study is conducted. The numerical analysis successfully demonstrates the applicability of the method for localizing mass perturbation and stiffness degradation under various EOC. This work contributes to proactive maintenance and damage detection for mechanical structures under varying conditions.

Harmonic State-Space Modeling and System Characteristic Analysis of Grid-Connected Inverter Parallel-Operation System Considering Asynchronous Carriers

Harmonic State-Space Modeling and System Characteristic Analysis of Grid-Connected Inverter Parallel-Operation System Considering Asynchronous Carriers

Transient Amplification of Broken Symmetry in Elastic Snap-Through.

A snap-through bifurcation occurs when a bistable structure loses one of its stable states and moves rapidly to the remaining state. For example, a buckled arch with symmetrically clamped ends can snap between an inverted and a natural state as the ends are released. A standard linear stability analysis suggests that the arch becomes unstable to asymmetric perturbations. Surprisingly, our experiments show that this is not always the case: symmetric transitions are also observed. Using experiments, numerics, and a toy model, we show that the symmetry of the transition depends on the rate at which the ends are released, with sufficiently fast loading leading to symmetric snap-through. Our toy model reveals that this behavior is caused by a region of the system's state space in which any initial asymmetry is amplified. The system may not enter this region when loaded fast (hence remaining symmetric), but will traverse it for some interval of time when loaded slowly, causing a transient amplification of asymmetry. Our toy model suggests that this behavior is not unique to snapping arches, but rather can be observed in dynamical systems where both a saddle-node and a pitchfork bifurcation occur in close proximity.

Infinite-dimensional reservoir computing

Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state–space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality when learning input/output systems in the class of generalized Barron functionals and measuring the error in a mean-squared sense.

Coordinated configuration of hybrid energy storage for electricity-hydrogen integrated energy system

This paper proposes an optimal coordinated configuration method of hybrid electricity and hydrogen storage for the electricity‑hydrogen integrated energy system (EH-ES) to promote the renewable energy source (RES) utilization and reduce the deployment cost. To simulate the practical operation of EH-ES, an energy hub framework with a discrete state space system matrix is designed to characterize the internal interconnection topology and quantify the steady-state production, storage, conversion and utilization processes of electricity and hydrogen energy carriers within EH-ES. A chronological operation simulation based electricity and hydrogen storage configuration model over a year-round time horizon is formulated to collaboratively optimize the capacity of the electrolyzer, fuel cell, battery energy storage (BES) and hydrogen storage tank. In this model, the power and capacity complementarity of BES and generalized hydrogen storage is fully exploited to compensate for power and energy imbalances, and additional capacity configuration of BES is considered to mitigate the drastic electrolytic power fluctuation. Besides, a typical source-load scenario reduction method based on K-means and ordered clustering is developed to avoid a large number of operation variables due to year-round hourly simulation. Comparative studies demonstrate that the proposed methodology can achieve the RES consumption rate of 98.873 % while the equipment investment cost can be reduced by 10.034 %.