State-space Dynamics Research Articles

Consistent spectral approximation of Koopman operators using resolvent compactification

Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L 2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.

An alternative EMG normalization method: Heterogeneous recurrence quantification analysis of isometric maximum voluntary contraction movements

This study presents an alternative method of normalizing Surface Electromyography (EMG) signals, which provide valuable insights into the musculoskeletal properties of the human body. Traditional normalization through the peak maximum voluntary contraction (MVC) method may result in variable outcomes due to its dependence on pre-processing steps such as smoothing and window length. To address this issue, the study standardizes the EMG pre-processing steps by using non-linear techniques, namely recurrence quantification analysis (RQA) and heterogeneous RQA (HRQA). RQA, specifically HRQA, functions as EMG feature extractors to explore the nonlinear dynamical system and state space trajectories of chaotic EMG signals. Principal component analysis (PCA) and kernel PCA (KPCA) were further applied to the RQA and HRQA results to obtain data with more information. The results show that the first principal components (PC1 and KPC1) of HRQA have a very strong relationship with peak MVC (ρ > |0.90|, p < 0.001), indicating their potential compared to the traditional EMG normalization technique for lower limb muscle groups. Additionally, the study investigated the dynamic differences among isometric MVC movements in lower limb muscle groups and found that the RQA and HRQA methods indicated a greater separation for detecting these differences compared to peak MVC. This study offers a valuable method for assessing the musculoskeletal characteristics of individuals who may have difficulty with activities like walking or providing maximum force during EMG data collection, such as those suffering from Parkinson's disease or walking disabilities. This enhances our understanding of their musculoskeletal properties.

A new model for compressor surge and stall control

This paper compares the bifurcations and closed-loop performances of two compressor models, Moore-Greitzer (MG) and a developed model based on MG (Shahriyari Khaleghi, SK). First, both models are linearized about two equilibrium points (pure surge and fully-developed rotating stall), and the perturbed state-space dynamics and input matrices are obtained. The compressor unstable regions are then identified using an eigenvalue and global bifurcation analysis. Furthermore, optimal LQR controllers are designed, and the performances of closed-loop systems are compared. The LQRs are designed to control the compression system near the peak pressure rise by suppressing surge or stall. Results reveal that if the initial operating point is in the positive slope region of the compressor characteristic and the initial amplitude of the disturbances is small, the LQR controller can stabilize the compressor in both models. However, when the disturbances are intensive, the two models respond differently: although the SK model damps a fair range of disturbances and predicts instability for excessively powerful disturbances, the MG model always damps them, even when extremely intense. Without a controller in the MG model, initial disturbances (even very large) can never grow and are always damped in the compressor’s negative slope region (obviously, the same applies to the controller). However, pending the amplitude of the disturbances (in the absence of a controller), the disturbances in the SK model may be damped or grow. The SK model can successfully control the instabilities if the disturbances are small. Nonetheless, the controller fails to dampen the instabilities for extreme disturbances, which is consistent with reality.

Frequency dependent whole-brain coactivation patterns analysis in Alzheimer's disease.

The brain in resting state has complex dynamic properties and shows frequency dependent characteristics. The frequency-dependent whole-brain dynamic changes of resting state across the scans have been ignored in Alzheimer's disease (AD). Coactivation pattern (CAP) analysis can identify different brain states. This paper aimed to investigate the dynamic characteristics of frequency dependent whole-brain CAPs in AD. We utilized a multiband CAP approach to model the state space and study brain dynamics in both AD and NC. The correlation between the dynamic characteristics and the subjects' clinical index was further analyzed. The results showed similar CAP patterns at different frequency bands, but the occurrence of patterns was different. In addition, CAPs associated with the default mode network (DMN) and the ventral/dorsal visual network (dorsal/ventral VN) were altered significantly between the AD and NC groups. This study also found the correlation between the altered dynamic characteristics of frequency dependent CAPs and the patients' clinical Mini-Mental State Examination assessment scale scores. This study revealed that while similar CAP spatial patterns appear in different frequency bands, their dynamic characteristics in subbands vary. In addition, delineating subbands was more helpful in distinguishing AD from NC in terms of CAP.

Dynamic phases induced by two-level system defects on driven qubits

Recent experimental evidences point to two-level defects, located in the oxides and on the interfaces of the Josephson junctions, as the major constituents of decoherence in superconducting qubits. How these defects affect the qubit evolution with the presence of external driving is less well understood since the semiclassical qubit-field coupling renders the Jaynes–Cummings model for qubit-defect coupling undiagonalizable. We analyze the decoherence dynamics in the continuous coherent state space induced by the driving and solve the master equation endowed with an extra decay-cladded driving term via a Fokker–Planck equation. The solutions for diffusion propagators as Gaussian distributions show four distinct dynamic phases: four types of convergence paths to limit cycles of varying radius by the distribution mean, which are determined by the competing external driving and the defect decays. The qubit trajectory resulted from these solutions is a super-Poissonian over displaced Fock states, which reduces to a Gibbs state of effective temperature decided by the defect at zero driving limit. Furthermore, the Poincare map shows the dependence of the rate of convergence on the initial state. In other words, the qubit evolution can serve as an indicator of the defect coupling strength through the variation of the driving strength as a parameter.

: A Semantics-Guided Safety Enhancement Framework for AI-Enabled Cyber-Physical Systems

Cyber-Physical Systems (CPSs) have been widely adopted in various industry domains to support many important tasks that impact our daily lives, such as automotive vehicles, robotics manufacturing, and energy systems. As Artificial Intelligence (AI) has demonstrated its promising abilities in diverse tasks like decision-making, prediction, and optimization, a growing number of CPSs adopt AI components in the loop to further extend their efficiency and performance. However, these modern AI-enabled CPSs have to tackle pivotal problems that the AI-enabled control systems might need to compensate the balance across <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiple operation requirements</i> and avoid possible defections in advance to safeguard human lives and properties. Modular redundancy and ensemble method are two widely adopted solutions in the traditional CPSs and AI communities to enhance the functionality and flexibility of a system. Nevertheless, there is a lack of deep understanding of the effectiveness of such ensemble design on AI-CPSs across diverse industrial applications. Considering the complexity of AI-CPSs, existing ensemble methods fall short of handling such huge state space and sophisticated system dynamics. Furthermore, an ideal control solution should consider the multiple system specifications in real-time and avoid erroneous behaviors beforehand. Such that, a new specification-oriented ensemble control system is of urgent need for AI-CPSs. In this paper, we propose <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathtt {SIEGE}$</tex-math></inline-formula> , a semantics-guided ensemble control framework to initiate an early exploratory study of ensemble methods on AI-CPSs and aim to construct an efficient, robust, and reliable control solution for multi-tasks AI-CPSs. We first utilize a semantic-based abstraction to decompose the large state space, capture the ongoing system status and predict future conditions in terms of the satisfaction of specifications. We propose a series of new semantics-aware ensemble strategies and an end-to-end Deep Reinforcement Learning (DRL) hierarchical ensemble method to improve the flexibility and reliability of the control systems. Our large-scale, comprehensive evaluations over five subject CPSs show that 1) the semantics abstraction can efficiently narrow the large state space and predict the semantics of incoming states, 2) our semantics-guided methods outperform state-of-the-art individual controllers and traditional ensemble methods, and 3) the DRL hierarchical ensemble approach shows promising capabilities to deliver a more robust, efficient, and safety-assured control system. To enable further research along this direction to build better AI-enabled CPS, we made all of the code and experimental results data publicly available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://sites.google.com/view/ai-cps-siege/home</uri> .

Learning strange attractors with reservoir systems

This paper shows that the celebrated embedding theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.

Using scaling-region distributions to select embedding parameters

Reconstructing state-space dynamics from scalar data using time-delay embedding requires choosing values for the delay τ and the dimension m. Both parameters are critical to the success of the procedure and neither is easy to formally validate. While embedding theorems do offer formal guidance for these choices, in practice one has to resort to heuristics, such as the average mutual information (AMI) method of Fraser & Swinney for τ or the false near neighbor (FNN) method of Kennel et al. for m. Best practice suggests an iterative approach: one of these heuristics is used to make a good first guess for the corresponding free parameter and then an “asymptotic invariant” approach is then used to firm up its value by, e.g., computing the correlation dimension or Lyapunov exponent for a range of values and looking for convergence. This process can be subjective, as these computations often involve finding, and fitting a line to, a scaling region in a plot: a process that is generally done by eye and is not immune to confirmation bias. Moreover, most of these heuristics do not provide confidence intervals, making it difficult to say what “convergence” is. Here, we propose an approach that automates the first step, removing the subjectivity, and formalizes the second, offering a statistical test for convergence. Our approach rests upon a recently developed method for automated scaling-region selection that includes confidence intervals on the results. We demonstrate this methodology by selecting values for the embedding dimension for several real and simulated dynamical systems. We compare these results to those produced by FNN and validate them against known results—e.g., of the correlation dimension—where these are available. We note that this method extends to any free parameter in the theory or practice of delay reconstruction.

Experimental evidence that rill-bed morphology is governed by emergent nonlinear spatial dynamics

Past experimental work found that rill erosion occurs mainly during rill formation in response to feedback between rill-flow hydraulics and rill-bed roughness, and that this feedback mechanism shapes rill beds into a succession of step-pool units that self-regulates sediment transport capacity of established rills. The search for clear regularities in the spatial distribution of step-pool units has been stymied by experimental rill-bed profiles exhibiting irregular fluctuating patterns of qualitative behavior. We hypothesized that the succession of step-pool units is governed by nonlinear-deterministic dynamics, which would explain observed irregular fluctuations. We tested this hypothesis with nonlinear time series analysis to reverse-engineer (reconstruct) state-space dynamics from fifteen experimental rill-bed profiles analyzed in previous work. Our results support this hypothesis for rill-bed profiles generated both in a controlled lab (flume) setting and in an in-situ hillside setting. The results provide experimental evidence that rill morphology is shaped endogenously by internal nonlinear hydrologic and soil processes rather than stochastically forced; and set a benchmark guiding specification and testing of new theoretical framings of rill-bed roughness in soil-erosion modeling. Finally, we applied echo state neural network machine learning to simulate reconstructed rill-bed dynamics so that morphological development could be forecasted out-of-sample.

Cyclic attractors of nonexpanding q-ary networks.

Discrete dynamical systems in which model components take on categorical values have been successfully applied to biological networks to study their global dynamic behavior. Boolean models in particular have been used extensively. However, multi-state models have also emerged as effective computational tools for the analysis of complex mechanisms underlying biological networks. Models in which variables assume more than two discrete states provide greater resolution, but this scheme introduces discontinuities. In particular, variables can increase or decrease by more than one unit in one time step. This can be corrected, without changing fixed points of the system, by applying an additional rule to each local activation function. On the other hand, if one is interested in cyclic attractors of their system, then this rule can potentially introduce new cyclic attractors that were not observed previously. This article makes some advancements in understanding the state space dynamics of multi-state network models with synchronous, sequential, or block-sequential update schedules and establishes conditions under which no new cyclic attractors are added to networks when the additional rule is applied. Our analytical results have the potential to be incorporated into modeling software and aid researchers in their analyses of biological multi-state networks.

Exact coherent states in plane Couette flow under spanwise wall oscillation

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods $({A_w}, T)$ . It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around ${T^{+}} \approx 100$ . The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

Using Nonlinear Vibroartrographic Parameters for Age-Related Changes Assessment in Knee Arthrokinematics.

Changes in articular surfaces can be associated with the aging process and as such may lead to quantitative and qualitative impairment of joint motion. This study is aiming to evaluate the age-related quality of the knee joint arthrokinematic motion using nonlinear parameters of the vibroarthrographic (VAG) signal. To analyse the age-related quality of the patellofemoral joint (PFJ), motion vibroarthrography was used. The data that were subject to analysis represent 220 participants divided into five age groups. The VAG signals were acquired during flexion/extension knee motion and described with the following nonlinear parameters: recurrence rate () and multi-scale entropy (). and decrease almost in a linear way with age (main effects of group ; means (): ; and . The post-hoc analysis showed that there were statistically significant differences () in all comparisons with the exception of the 5th–6th life decade. For , statistically significant differences () occurred for: 3rd–7th, 4th–7th, 5th–7th and 6th life decades. Our results imply that degenerative age-related changes are associated with lower repeatability, greater heterogeneity in state space dynamics, and greater regularity in the time domain of VAG signal. In comparison with linear VAG measures, our results provide additional information about the nature of changes of the vibration dynamics of PFJ motion with age.

Autodifferentiable Ensemble Kalman Filters

Data assimilation is concerned with sequentially estimating a temporally evolving state. This task, which arises in a wide range of scientific and engineering applications, is particularly challenging when the state is high-dimensional and the state-space dynamics are unknown. This paper introduces a machine learning framework for learning dynamical systems in data assimilation. Our auto-differentiable ensemble Kalman filters (AD-EnKFs) blend ensemble Kalman filters for state recovery with machine learning tools for learning the dynamics. In doing so, AD-EnKFs leverage the ability of ensemble Kalman filters to scale to high-dimensional states and the power of automatic differentiation to train high-dimensional surrogate models for the dynamics. Numerical results using the Lorenz-96 model show that AD-EnKFs outperform existing methods that use expectation-maximization or particle filters to merge data assimilation and machine learning. In addition, AD-EnKFs are easy to implement and require minimal tuning.

Verification framework for control theory of aircraft

AbstractA control system verification framework is presented for unmanned aerial vehicles using theorem proving. The framework’s aim is to set out a procedure for proving that the mathematically designed control system of the aircraft satisfies robustness requirements to ensure safe performance under varying environmental conditions. Extensive mathematical derivations, which have formerly been carried out manually, are checked for their correctness on a computer. To illustrate the procedures, a higher-order logic interactive theorem-prover and an automated theorem-prover are utilised to formally verify a nonlinear attitude control system of a generic multi-rotor UAV over a stability domain within the dynamical state space of the drone. Further benefits of the procedures are that some of the resulting methods can be implemented onboard the aircraft to detect when its controller breaches its flight envelope limits due to severe weather conditions or actuator/sensor malfunction. Such a detection procedure can be used to advise the remote pilot, or an onboard intelligent agent, to decide on some alterations of the planned flight path or to perform emergency landing.

Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

AbstractWe consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems.In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

Distributed Model Predictive Control of Time-Delay Systems

In this paper, a new approach to distributed MPC of linear interconnected time-delay systems is proposed. The general case of time-delay in both the states and the control inputs is considered. The approach includes representation of the interconnected time-delay dynamics in the augmented state space and reformulation of the centralized MPC problem into a Quadratic Programming framework. The latter this is solved distributedly by using the dual gradient method for optimization. The suggested approach would be appropriate for embedded distributed MPC and is illustrated on a simulation example of two interconnected time-delay systems.

A Detailed Full-Order Discrete-Time Modeling and Stability Prediction of the Single-Phase Dual Active Bridge DC-DC Converter

The standard methodology to obtain the model of a power electronic converter is achieved by averaging the state-space dynamics of the converter’s state variables. But the average of the transformer current is null over a switching cycle in the resonant dc-dc converter. Therefore, the conventional method is not suitable for resonant converters, including the phase-shifted bidirectional dual active bridge (PSBDAB) converter. The two-time scale discrete-type models can resolve the problem associated with the standard state-space averaging methodology. The time-scale segregates the dynamics of the PSBDAB converter into fast and slow state variables, which can be modeled separately and eases the analysis of the PSBDAB converter. The effect of the core-loss of the inductor, dead-time of the semiconductor devices, output filter capacitor’s equivalent series resistance, semiconductor on-resistance, and the transformer copper loss components are included in the model to improve its steady-state and dynamics characteristics. Moreover, the stability analysis using a bifurcation diagram is carried out for the digitally controlled closed-loop of the system. Furthermore, the critical gain for the stable region with variations in the circuit parameters like load resistance, circuit equivalent inductance, and voltage demand is extensively studied. The modeling and stability analysis is validated in the simulation and experimental setup. The results verify that the proposed method accurately predicts the stable region with variations in the system circuit parameters. Thus this study provides a guide to select and tune the controller parameter to ensure the converter operates within the boundaries of the stable region.

Traffic Control Prediction Design Based on Fuzzy Logic and Lyapunov Approaches to Improve the Performance of Road Intersection

Due to the increasing use of private cars for urbanization and urban transport, the travel time of urban transportation is increasing. People spend a lot of time in the streets, and the queue length of waiting increases accordingly; this has direct effects on fuel consumption too. Traffic flow forecasts and traffic light schedules were studied separately in the urban traffic system. This paper presents a new stable TS (Takagi–Sugeno) fuzzy controller for urban traffic. The state-space dynamics are utilized to formulate both the vehicle’s average waiting time at an isolated intersection and the length of queues. A fuzzy intelligent controller is designed for light control based upon the length of the queue, and eventually, the system’s stability is proved using the Lyapunov theorem. Moreover, the input variables are the length of queue and number of input or output vehicles from each lane. The simulation results describe the appearance of the proposed controller. An illustrative example is also given to show the proposed method’s effectiveness; the suggested method is more efficient than both the conventional fuzzy traffic controllers and the fixed time controller.

Optimum control of purpose activities in the area of quality based on a linear model of dynamics in state space

The paper considers the optimization problem formulation at goal-setting in the field of quality on the basis of a dynamics verified linear model in the state space using stepwise and linear laws of enterprise activity management. The classical quadratic functional to find optimal solutions for goal-setting was used. The direct connection of these functional components with the Taguchi loss function for product quality indicators has been substantiated, and the function scope application on the managing purposeful activities process has been expanded. The optimization parameters are determined, which are the amplification factors of the control actions. These actions ensure the actual attainability of the set goals in the field of quality during the time planned period. The optimal solutions search was carried out on the example of the textile and light industry CJSC “Salyut” (St. Petersburg). The functional minimum value in case of stepwise control law is more than 14 times greater than with a linear law. This fact found under the condition of equal unit costs for quality losses and control caused by their deviations from the nominal values. The results obtained can be used for a reasonable choice of the enterprise management law in the goal-setting in the field of quality.

Multiframe Evolving Dynamic Functional Network Connectivity Motifs (Evodfncs) from Continuity-Preserving Planar Embedding.

The study of brain network connectivity as a time-varying property began relatively recently and to date has remained primarily concerned with capturing a handful of discrete static states that characterize connectivity as measured on a timescale shorter than that of the full scan. Capturing group- level representations of temporally evolving patterns of connectivity is a challenging and important next step in fully leveraging the information available in large resting state functional magnetic resonance imaging (rs-fMRI) studies. We introduce a flexible, extensible data-driven framework for the identification of group-level multiframe (movie-style) dynamic functional network connectivity (dFNC) states. Our approach employs uniform manifold approximation and embedding (UMAP) to produce a planar embedding of the high-dimensional whole-brain connectivity dynamics that preserves important features, such as trajectory continuity, characterizing dynamics in the native high dimensional state space. The method is validated in application to a large rs- fMRI study of schizophrenia where it extracts naturalistic fluidly-varying connectivity motifs that differ between schizophrenia patients (SZs) and healthy controls (HC).Functional Magnetic Resonance Imaging, Functional Network Connectivity, Dynamic Functional Network Connectivity, Schizophrenia.