Time-invariant Model Research Articles

Linear Time-Invariant Approximations of Nonlinear Time-Periodic Systems

This paper discusses the development of a numerical method for the approximation of the nonlinear time-periodic rotorcraft flight dynamics with higher order linear time-invariant (LTI) models. The method relies on a per-rotor revolution perturbation scheme, which is of particular importance for the linearization of simulation models that do not allow for per-time-step perturbations, and for those output measures that necessitate the solution of partial differential equations and thus require several time steps to be computed. The paper demonstrates the application of the proposed methodology to obtain high-order LTI models capable of predicting vibrations for a generic utility helicopter. Simulations are used to validate the response of the linearized models against those from nonlinear simulations and from competing approaches in the literature. The proposed method is shown to predict accurately the nonlinear response for the case shown and for small amplitude maneuvers. Frequency-domain validation is also performed to compare the linear models derived with the proposed method with those obtained with harmonic decomposition, a competing approach based on a per-time-step perturbation scheme. Interestingly, the proposed algorithm yields nearly identical numerical results compared to harmonic decomposition, suggesting that the two methods are in fact equivalent but rely on different formulations.

Model Predictive Control in Partially Observable Multi-Modal Discrete Environments

Autonomous systems operate in environments that can be observed only through noisy measurements. Thus, controllers should compute actions based on their beliefs about the surroundings. In these settings, we design a Model Predictive Controller (MPC) based on a continuous-state Linear Time-Invariant (LTI) system model operating in a discrete-state environment described by a Hidden Markov Model (HMM). Environment constraints are modeled as chance constraints and environment observations can be asynchronous with system state measurements and controller updates. We show how to approximate the solution of the MPC problem defined over the space of feedback policies by optimizing over a trajectory tree, where each branch is associated with an environment measurement. The proposed approach guarantees chance constraint satisfaction and recursive feasibility. Finally, we test the proposed strategy on navigation examples in partially observable environments, where the proposed MPC guarantees chance constraint satisfaction.

Assessment of COVID-19 Hospitalization Forecasts from a Simplified SIR Model

We propose the SH model, a simplified version of the well-known SIR compartmental model of infectious diseases. With optimized parameters and initial conditions, this time-invariant two-parameter two-dimensional model is able to fit COVID-19 hospitalization data over several months with high accuracy (e.g., the root relative squared error is below 10% for Belgium over the period from 2020-03-15 to 2020-07-15). Moreover, we observed that, when the model is trained on a suitable three-week period around the hospitalization peak for Belgium, it forecasts the subsequent two months with mean absolute percentage error (MAPE) under 4%. We repeated the experiment for each French department and found 14 of them where the MAPE was below 20%. However, when the model is trained in the increase phase, it is less successful at forecasting the subsequent evolution.

On off-line and on-line Bayesian filtering for uncertainty quantification of structural deterioration

Abstract Data-informed predictive maintenance planning largely relies on stochastic deterioration models. Monitoring information can be utilized to update sequentially the knowledge on model parameters. In this context, on-line (recursive) Bayesian filtering algorithms typically fail to properly quantify the full posterior uncertainty of time-invariant model parameters. Off-line (batch) algorithms are—in principle—better suited for the uncertainty quantification task, yet they are computationally prohibitive in sequential settings. In this work, we adapt and investigate selected Bayesian filters for parameter estimation: an on-line particle filter, an on-line iterated batch importance sampling filter, which performs Markov Chain Monte Carlo (MCMC) move steps, and an off-line MCMC-based sequential Monte Carlo filter. A Gaussian mixture model approximates the posterior distribution within the resampling process in all three filters. Two numerical examples provide the basis for a comparative assessment. The first example considers a low-dimensional, nonlinear, non-Gaussian probabilistic fatigue crack growth model that is updated with sequential monitoring measurements. The second high-dimensional, linear, Gaussian example employs a random field to model corrosion deterioration across a beam, which is updated with sequential sensor measurements. The numerical investigations provide insights into the performance of off-line and on-line filters in terms of the accuracy of posterior estimates and the computational cost, when applied to problems of different nature, increasing dimensionality and varying sensor information amount. Importantly, they show that a tailored implementation of the on-line particle filter proves competitive with the computationally demanding MCMC-based filters. Suggestions on the choice of the appropriate method in function of problem characteristics are provided.

Order of Occurrence of COVID-19 Symptoms.

COVID-19 symptoms change after onset-some show early, others later. This article examines whether the order of occurrence of symptoms can improve diagnosis of COVID-19 before test results are available. In total, 483 individuals who completed a COVID-19 test were recruited through Listservs. Participants then completed an online survey regarding their symptoms and test results. The order of symptoms was set according to (a) whether the participant had a "history of the symptom" due to a prior condition; and (b) whether the symptom "occurred first," or prior to, other symptoms of COVID-19. Two LASSO (Least Absolute Shrinkage and Selection Operator) regression models were developed. The first model, referred to as "time-invariant," used demographics and symptoms but not the order of symptom occurrence. The second model, referred to as "time-sensitive," used the same data set but included the order of symptom occurrence. The average cross-validated area under the receiver operating characteristic (AROC) curve for the time-invariant model was 0.784. The time-sensitive model had an AROC curve of 0.799. The difference between the 2 accuracy levels was statistically significant (α < .05). The order of symptom occurrence made a statistically significant, but small, improvement in the accuracy of the diagnosis of COVID-19.

Modal-Based Anisotropy Early Warning in Wind Turbine Rotor

Subspace-based fault detection methods are widely used for linear time-invariant systems. For linear time-periodic systems, those methods cannot be theoretically used, due to the intrinsic assumptions associated with those methods in the context of linear time-invariant models. Based on the approximation of time-periodic systems as time-invariant ones, those methods can still be applied and adapted to perform change detection for time-periodic systems, through a Gaussian residual built upon the identified modal parameters and their estimated variances. The proposed method is tested and validated on a small numerical model of a rotating wind turbine, with detection and isolation of a blade stiffness reduction leading to rotor anisotropy.

Using structured low-rank tensors for multilinear modeling of building systems

The paper proposes a low-rank structured parameter identification method for multilinear models. Multilinear models extend the class of linear time-invariant models and can depict more complex dynamics. Tensor representations of these models keep the dimensionality but provide very efficient storage and computation methods of the models by applying decomposition and normalization procedures. The proposed parameter identification method is an automated grey box approach, where no manual modeling is required. A pre-structuring process reduces the parameter identification problem and converts it to a sparse representation to make it applicable to large scale applications while preserving the interpretability of the parameters as property of the normalized multilinear models. An application example for anomaly detection of building systems is given with simulation data for the HVAC system of a seminar room.

Electric Autonomous Mobility-on-Demand: Joint Optimization of Routing and Charging Infrastructure Siting

The advent of vehicle autonomy, connectivity and electric powertrains is expected to enable the deployment of Autonomous Mobility-on-Demand systems. Crucially, the routing and charging activities of these fleets are impacted by the design of the individual vehicles and the surrounding charging infrastructure which, in turn, should be designed to account for the intended fleet operation. This paper presents a modeling and optimization framework where we optimize the activities of the fleet jointly with the placement of the charging infrastructure. We adopt a mesoscopic planning perspective and devise a time-invariant model of the fleet activities in terms of routes and charging patterns, explicitly capturing the state of charge of the vehicles by resampling the road network as a digraph with iso-energy arcs. Then, we cast the problem as a mixed-integer linear program that guarantees global optimality and can be solved in less than 10 min. Finally, we showcase two case studies with real-world taxi data in Manhattan, NYC: The first one captures the optimal trade-off between charging infrastructure prevalence and the empty-mileage driven by the fleet. We observe that jointly optimizing the infrastructure sitting significantly outperforms heuristic placement policies, and that increasing the number of stations is beneficial only up to a certain point. The second case focuses on vehicle design and shows that deploying vehicles equipped with a smaller battery results in the lowest energy consumption: Although necessitating more trips to the charging stations, such fleets require about 12% less energy than the vehicles with a larger battery capacity.

On the herdability of linear time-invariant systems with special topological structures

In this paper we investigate the herdability property, namely the capability of a system to be driven towards the (interior of the) positive orthant, for linear time-invariant state space models. Herdability of certain matrix pairs (A,B), where A is the adjacency matrix of a multi-agent network, and B a selection matrix that singles out a subset of the agents (the “network leaders”), is explored. The cases when the graph associated with A, G(A), is directed and clustering balanced (in particular, structurally balanced), or it has a tree topology and there is a single leader, are investigated.

Improving multi-decadal coastal shoreline change predictions by including model parameter non-stationarity

Our ability to predict sandy shoreline evolution resulting from future changes in regional wave climates is critical for the sustainable management of coastlines worldwide. To this end, the present generation of simple and efficient semi-empirical shoreline change models have shown good skill at predicting shoreline changes from seasons up to several years at a number of diverse sites around the world. However, a key limitation of these existing approaches is that they rely on time-invariant model parameters, and assume that beaches will evolve within constrained envelopes of variability based on past observations. This raises an interesting challenge because the expected future variability in key meteocean and hydrodynamic drivers of shoreline change are likely to violate this ‘stationary’ approach to longer-term shoreline change prediction. Using a newly available, multi-decadal (28-year) dataset of satellite-derived shorelines at the Gold Coast, Australia, this contribution presents the first attempt to improve multi-decadal shoreline change predictions by allowing the magnitude of the shoreline model parameters to vary in time. A data assimilation technique (Ensemble Kalman Filter, EnKF) embedded within the well-established ShoreFor shoreline change model is first applied to a 14-year training period of approximately fortnightly shoreline observations, to explore temporal variability in model parameters. Then, the magnitudes of these observed non-stationary parameters are modelled as a function of selected wave climate covariates, representing the underlying seasonal to interannual variability in wave forcing. These modelled time-varying parameters are then incorporated into the shoreline change model and tested over the complete 28-year dataset. This new inclusion of non-stationary model parameters that are directly modelled as a function of the underlying wave forcing and corresponding time scales of beach response, is shown to outperform the multi-decadal predictions obtained by applying the conventional stationary approach (RMSEnon-stationary = 11.1 m; RMSEstationary = 254.3 m). Based on these results, it is proposed that a non-stationary approach to shoreline change modelling can reduce the uncertainty associated with the misspecification of physical processes driving shoreline change and should be considered for future shoreline change predictions.

Simplified complex-valued modal model for operating wind turbines through aerodynamic decoupling and multi-blade coordinate transformation

An efficient modelling methodology for steady-state operating wind turbines is proposed, combining aerodynamic decoupling, multi-blade coordinate transformation, and modal reduction. This leads to a complex-valued, reduced-order modal model for prediction of dynamic responses of the tower-rotor-blade wind turbine system, considering rotor rotation, blade flexibility, and vibration coupling between rotor and tower. A fully coupled finite element model was first developed, with aerodynamic forces linearised and expressed as a viscous aerodynamic damping matrix. A time-invariant state space model is formed using multi-blade coordinate transformation, allowing standard modal analysis. The complex-valued eigenvalues and mode shapes were obtained, and it is shown that the operating wind turbine modes exhibit a combination of tower and blade vibrations. Various degrees of modal reduction are applied to the state space model to obtain a modal model with fewer degrees of freedom, whose performance was evaluated in the time and frequency domain for operating wind turbines in normal condition. The displacement and stress responses are in close agreement with those of the fully coupled model with the first 21 modes included. The model already performs well with 8 modes considered to capture relevant fundamental frequency peaks. This allows significantly reduced computational effort and can be particularly beneficial for fatigue prediction, reliability analysis, and structural identification.

Data-Driven Iterative Learning Control of Nonlinear Systems by Adaptive Model Matching

Iterative learning control (ILC) has proven successful in the industry for enhancing tracking performance of repetitive tasks. The high accuracy and fast convergence of ILC algorithms hinge on: 1) the knowledge of the system model and 2) an effective learning algorithm. For general industrial systems with nonlinear dynamics, this raises technical challenges because acquiring a nonlinear dynamical model or several linearized models at different operating points may be difficult and costly. It is also nontrivial to determine the learning algorithm for the complex model obtained. To address these challenges, this article proposes a novel data-driven ILC algorithm for single-input-single-output nonlinear systems. Without explicit nonlinear models, our algorithm treats the nonlinear system as an unknown linear time-varying system linearized on a specified input–output (I/O) trajectory in each ILC iteration. A linearly parameterized time-varying adaptive filter is constructed in each ILC iteration so that, when cascading with the nonlinear plant, the I/O dynamics around the specified trajectory follow a linear time-invariant reference model. The ILC error trajectory is then filtered by the adaptive time-varying filter, which represents the inverse dynamics with a bandwidth specified by the reference model, to render fast convergence. The benefits of the proposed data-driven algorithm is demonstrated by comparing to a gradient and Hessian-based data-driven ILC algorithm on a prototypical two-degree-of-freedom nonlinear pendulum.

Is Intermittent Control the Source of the Non-Linear Oscillatory Component (0.2-2Hz) in Human Balance Control?

To explain the 0.2-2Hz oscillation in human balance. Oscillation (0.2-2 Hz) in the control signal (ankle moment) is sustained independently of external disturbances and exaggerated in Parkinson's disease. Does resonance or limit cycles in the neurophysiological feedback loop cause this oscillation? We investigate two linear (non-predictive, predictive) and one non-linear (intermittent-predictive) control model (NPC, PC, IPC). Fourteen healthy participants, strapped to an actuated single segment robot with dynamics of upright standing, used natural haptic-visual feedback and myoelectric control signals from lower leg muscles to maintain balance. An input disturbance applied stepwise changes in external force. A linear time invariant model (ARX) extracted the delayed component of the control signal related linearly to the disturbance, leaving the remaining, larger, oscillatory non-linear component. We optimized model parameters and noise (observation, motor) to replicate concurrently (i) estimated-delay, (ii) time-series of the linear component, and (iii) magnitude-frequency spectrum and transient magnitude response of the non-linear component. Results (mean±S.D., p<0.05): NPC produced estimated delays (0.116±0.03s) significantly lower than experiment (0.145±0.04s). Overall fit (i)-(iii) was (79±7%, 83±7%, 84±6% for NPC, PC, IPC). IPC required little or no noise. Mean frequency of experimental oscillation (0.99±0.16 Hz) correlated trial by trial with closed loop resonant frequency (fres), not limit cycles, nor sampling rate. NPC (0.36±0.08Hz) and PC (0.86±0.4Hz) showed fres significantly lower than IPC (0.98±0.2Hz). Human balance control requires short-term prediction. IPC mechanisms (prediction error, threshold related sampling, sequential re-initialization of open-loop predictive control) explain resonant gain without uncontrolled oscillation for healthy balance.

Fast and Precise Generic Model for Position-Based Trajectory Prediction of Inland Waterway Vessels

Vessel motion simulation as well as model-based accurate trajectory prediction of vessels require accurate models with respect to related dynamic properties. The ability to predict vessel’s trajectory behaviors will become relevant in the case of future autonomous navigation of vessels to predict the behavior of others. The definition of models or parameters can be realized via first principles or by using experimental modeling methods leading to a time invariant or variant model. Existing hydrodynamical modeling approaches are based on mathematical approaches, which use parameters like mass, hydrodynamic forces, wind velocity, depth under the keel, loading parameters, etc. So, determining a dynamic vessel’s model is a complex task, since the model is vessel-specific. For collision avoidance of autonomous or assisted vessels, the trajectory prediction of encountering other vessels is especially required. It is not possible to use complex hydrodynamical models of encountering vessels online due to missing required information/measurements. Even existing deep learning approaches provide better predictions, but are still insufficient for collision avoidance in the case of strong dynamical changes, since the considered input sequences are long. Due to long input sequences, the model does not adapt to strong dynamical changes. In this work, a simple parameter-based approach is developed to predict the intended behavior using the last seconds of the measured position variables. The idea is to globally identify the model parameters of the vessel, which remains constant for the situation, and additionally two parameters for local adaptation, which adapt at every updated input sequence. Typically parameters like rudder angle, wind velocities, and water current affect the behavior of vessels. The introduced approach works with a sliding window approach for which, after identification of the global system, local values are identified based on the last 80 measurements of the vessels. A trajectory prediction (assuming no additional rudder-based maneuvering) is realized for the prediction horizon of 180 s. To confirm the robustness of the new approach, real AIS/GPS-based measurements from a German research inland vessel for different scenarios and sailing conditions including ‘loaded’ and ‘empty’ sailing cases are used. Furthermore, additional results are shown for position data information of different sample rates.

Determining appropriate input excitation for model identification of a continuous bio-process

This manuscript addresses the problem of determining input excitation for data driven model identification appropriate for cell culture bio-processes in general, and for an industrial bioreactor used for the production of monoclonal antibodies, in particular. The design space is set up to give us the operating parameters for the key objective of demonstrating the feasibility of using far more perturbations than typically done in bio process identification, although significantly less than other applications, to yield data rich enough for the purpose of data driven modelling (and subsequently, control). A proprietary mechanistic model developed by Sartorius for their Cellca cell line is first introduced to serve as a test bed, based on AMBR 250® (Sartorius registered trademark for integrated high throughput bioreactor systems). Subsequently, this test bed is used to address the question of determining the frequency of input perturbation sufficient to identify a data driven dynamic model. To this end, the test bed is used to generate data at various frequencies and a linear time invariant model identified. The predictive capability of the identified model is used to ascertain the frequency of changes in data generation such that the changes are acceptable from a biological standpoint, and yet generate sufficiently rich data. In particular, a frequency of perturbations at once every three days is found to balance these tradeoffs for the monoclonal antibody process under consideration. The results from the manuscript are meaningful both from a specific results standpoint (as illustrated by subsequent adoption by Sartorius), but also by providing a mechanism to ascertain such information for other bio-processes.

Oil shocks and global economy

This paper analyzes how global economic activity reacts to shocks in the crude oil market, allowing that such reactions may change over time by using a Time-Varying Parameter Vector Autoregression model. Our findings show that an unanticipated disruption in oil supply leads to a relevant decline in global economic activity at any period of time considered (1973:08–2019:06) and at any response horizon, with impacts being relatively similar over time. An unexpected expansion in aggregate demand meaningfully boots global economic activity at any period of time and at any response horizon. In contrast, an unexpected oil-specific demand increase only has a short-lived substantial effect on global economic activity for the shocks produced in the early 2000s, the global financial crisis and the shocks occurred in the last years of the 2010s. These findings are in line with those of Baumeister and Hamilton (2019) who, using a time-invariant model, found that oil supply shocks reduce global economic activity, but oil consumption-demand shocks do not.

Prediction of charge-discharge behavior and state of charge estimation for tri-electrode rechargeable zinc-air flow batteries

An effective state of charge (SOC) estimation is essential for the development of a battery management system. At present, the provision of a SOC estimation for zinc-air flow batteries (ZAFBs) is still at its early stage of development. This work sets out to develop the SOC estimator for ZAFBs. The estimator is based-on a linear parameter varying (LPV) model integrated with an extended Kalman filter (EKF). The LPV model is constructed from multiple linear time-invariant (LTI) models with battery current and SOC as scheduling parameters. It is observed that the response data for ZAFBs have an exceptionally flat profile related to SOC change; dynamic differentiation only occurs when SOC is almost depleted. For this reason, the estimation of SOC converges to the true value when SOC is near depletion. In this work, it is shown that by appropriate tuning, the SOC estimation performance of the LPV model combined with EKF performs well as the absolute errors of soc. estimation lie under 2 % after true SOC convergence. The LPV model with the EKF algorithm is also compared with the Luenberger observer (LO). The proposed estimator can surpass the LO estimator. Overall, this SOC estimator provides a systematic way to fulfill the requirements of a battery management system.

Stability Assessment of Modular Multilevel Converters Based on Linear Time-Periodic Theory: Time-Domain vs. Frequency-Domain

The periodic characteristic of modular multilevel converter (MMC), which originates from the multi-harmonic couplings among internal dynamics, results in challenges to develop the linear time-invariant (LTI) model of MMC. The linearization of MMC is inherently around time-periodic equilibrium trajectory. In this paper, two different methods to assess small-signal stability of MMC are compared. Both are based on the linear time-periodic (LTP) systems theory, but one is in the time domain (TD) and the other is in the frequency domain (FD). The LTP model of MMC is derived from the linearized arm averaged model. Then, an LTI model for the LTP system is constructed in the FD through the harmonic state-space method. Small-signal stability analysis based on Floquet theory and generalized Nyquist criterion, that accompany TD and FD, are introduced and compared. It is demonstrated that models developed in both domains can effectively capture the small-signal behavior of MMC. Third and higher-order harmonic components of the states have significant effect on the impedance-based stability analysis of MMC around mid-frequency range. Analytical results are validated by electromagnetic transient simulations based on the MMC aggregated model.

Bayesian dynamic programming approach for tracking time-varying model properties in SHM

Structural health monitoring (SHM) quite often involves continually tracking the temporal variation of some properties of interest, where statistical information about variations under normal situation can be obtained and potential anomalies may be detected for further attention. Structural-related properties such as natural frequency and stiffness often need to be identified with a physics-based model that relates them to measured data. Conventional approaches empirically divide data into non-overlapping segments with equal lengths and identify the model parameters within each segment based on a time-invariant model. Potentially time-varying model properties are then investigated based on the variation of identified results from one data segment to another. Challenges do exist, e.g., how to choose segment length to balance modeling error and identification accuracy, how to set criterion for anomaly detection, etc., which should be addressed with proper extraction of probabilistic information from data. In this work, the SHM problem is formulated as a Bayesian model selection problem, where the data can be ‘partitioned’ in an arbitrary manner, whose optimal choice, and hence points of significant change, are determined together with the model inference process by maximizing the probabilistic evidence supported by data. An efficient algorithm based on dynamic programming is proposed to determine the optimal partitioning and associated piecewise-constant properties, which is otherwise computationally prohibitive. The methodology is applied to tracking modal properties, e.g., natural frequency and damping ratio, of structures using output-only ambient vibration data. It is investigated with synthetic data and then applied to field data of a tall building during a typhoon event.

Energy Management in Buildings: Lessons Learnt for Modeling and Advanced Control Design

This paper presents a comparative analysis of different modeling and control techniques that can be used to tackle the energy efficiency and management problems in buildings. Multiple resources are considered, from generation to storage, distribution and delivery. In particular, it is shown what are the real needs and advantages of adopting different techniques, based on different applications, type of buildings, boundary conditions. This contribution is based widely on the experience performed by the authors in the recent years in dealing with existing residential, commercial and tertiary filed buildings, with application ranging from local temperature control up to smart grids where buildings are seen as an active node of the grid thanks to their ability to shape the thermal and electrical profile in real time. As for control models, a wide range of modeling techniques are here investigated and compared, from linear time-invariant models, to time-varying, to nonlinear ones. Similarly, control techniques include adaptive ones and real-time predictive ones.