Nonlinear State Variables Research Articles

Low conservative stability criteria for discrete-time Lur’e systems with sector and slope constrained nonlinearities

Lur’e system is an important nonlinear system in engineering, which is composed of a feedback connection of a linear dynamic system and a nonlinear system that satisfies the sector constraint. In this paper, for nominal and uncertain discrete Lur’e systems with time-varying delays, the absolute and robustly absolute stability are studied in detail with the nonlinearity satisfying the sector and slope constraints. Firstly, an augmented Lyapunov–Krasovskii functional (LKF) is designed, in which some augmented vectors are selected to increase the coupling information between the delay intervals and other system state variables. At the same time, some effective integral terms of nonlinear state variables are extended in the LKF according to the characteristics of sector constraint and slope constraint. Secondly, a general lemma of summation inequality for free-weighted matrices is given, which adds some coupling information between vectors and extends the search range for solving linear matrix inequalities (LMIs). Thirdly, by synthesizing the general summation inequality and the newly constructed LKF, different absolute stability criteria are derived respectively, and the absolute stability criteria are extended to robustly absolute stability cases with uncertain parameters. The stability criteria reduce the conservatism of some previously proposed results. Finally, some common numerical examples and simulations are used to verify the results of this paper.

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

Combination of physics‐based and data‐driven modeling for nonlinear structural seismic response prediction through deep residual learning

AbstractDespite great progress in seeking accurate numerical approximator to nonlinear structural seismic response prediction using deep learning approaches, tedious training process and large volume of structural response data under earthquakes for training and validation are often prohibitively accessible. In our methodology, the main innovation can be seen in the incorporation of deep neural networks (DNNs) into a classical numerical integration method by using a hybridized integration time‐stepper. In this way, the linear physics information of the structure and the obscure nonlinear dynamics are smoothly combined. We propose to use residual network (ResNet) to learn time‐stepping schemes specifically for the nonlinear state variables of the system. Our Physics‐DNN hybridized integration (PDHI) time‐stepping scheme provides important advantages over current pure data‐driven approaches, including (i) a flexible framework incorporating known time‐invariant physics information, (ii) requirement of structural seismic response data being circumvented by simple short bursts of trajectories collected from underlying nonlinear components, and (iii) efficiency in training and validation process. Besides, our results indicate that a simple feedforward or convolutional architecture outperforms recurrent networks to fulfill the requirement of prediction accuracy as well as long‐range memory in structural dynamic analysis. Several numerical and experimental examples are presented to demonstrate the performance of the method.

Enabling in-time prognostics with surrogate modeling through physics-enhanced Dynamic Mode Decomposition method

Computational models provide essential quantitative tools for assessing and predicting the health and performance of physical systems. However, high-fidelity models are rarely used in real-time operations or large optimization loops, due to their time-intensive nature. A common approach to improving computational efficiency of prognosis is to employ surrogate models. Such models can significantly decrease computation time for some accuracy loss. In this context, use of Dynamic Mode Decomposition (DMD) is proposed to generate surrogate models for lithium-ion (Li-ion) battery discharge. DMD has been suggested and used successfully in the area of fluid dynamics for over a decade, but it has not been applied to the Prognostics and Health Management domain, where farahead prediction of nonlinear behavior is crucial to propagate faults or predict Remaining Useful Life (RUL). For Li-ion battery health management, the standard application of DMD using only the observable quantities of interest was unable to capture the nonlinear discharge of batteries exhibited in lab testing. A potential solution was found by implementing Koopman theory, which considers the dynamics of nonlinear systems. Koopman theory provides a mechanism to trade-off low dimensional nonlinear models with high-dimensional linear ones in a DMD framework, by augmenting nonlinear state variables into the system representation. For battery health management, we augmented the observable variables with the hidden states of a higher-fidelity physics model to build the DMD surrogate. In comparison to a high-fidelity model, the surrogate improved computational efficiency with only a minimal loss of accuracy, and enabled long-term prognostics horizons. A generalized method for this was implemented in the ProgPy python packages.

Integrated nonlinear suboptimal control‐and‐estimator based on the state‐dependent differential Riccati equation approach

AbstractThis paper is devoted to the theory and application of a combined nonlinear suboptimal control and estimator based on the state‐dependent differential Riccati equation (SDDRE), aiming finite‐time tracking of manipulators that possess flexible joints. Though measuring the angular positions of links is possible for flexible joint manipulators (FJMs), it is not feasible to measure the changes in the position angles of actuators. In addition, the existence of measurement noise and the influence of unknown workspace disturbance are always unavoidable. Therefore, combining an estimator with a controller could result in full‐state feedback for the system. Designing the proposed integrated estimator algorithm includes two parts: The first part is the theoretical extension of the SDDRE approach expressed with nonlinear state variables and inputs of the system in a finite‐time horizon. In the second part, enhancing and upgrading software and hardware interface, the proposed procedure for FJMs is modeled and simulated. Then, the experimental implementation is applied in the online closed‐loop form.

Bayesian learning of stochastic dynamical models

A new methodology for rigorous Bayesian learning of high-dimensional stochastic dynamical models is developed. The methodology performs parallelized computation of marginal likelihoods for multiple candidate models, integrating over all state variable and parameter values, and enabling a principled Bayesian update of model distributions. This is accomplished by leveraging the dynamically orthogonal (DO) evolution equations for uncertainty prediction in a dynamic stochastic subspace and the Gaussian Mixture Model-DO filter for inference of nonlinear state variables and parameters, using reduced-dimension state augmentation to accommodate models featuring uncertain parameters. Overall, the joint Bayesian inference of the state, model equations, geometry, boundary conditions, and initial conditions is performed. Results are exemplified using two high-dimensional, nonlinear simulated fluid and ocean systems. For the first, limited measurements of fluid flow downstream of an obstacle are used to perform joint inference of the obstacle’s shape, the Reynolds number, and the O(105) fluid velocity state variables. For the second, limited measurements of the concentration of a microorganism advected by an uncertain flow are used to perform joint inference of the microorganism’s reaction equation and the O(105) microorganism concentration and ocean velocity state variables. When the observations are sufficiently informative about the learning objectives, we find that our posterior model probabilities correctly identify either the true model or the most plausible models, even in cases where a human would be challenged to do the same.

Learning Flatness-Based Controller Using Neural Networks

Abstract This paper presents a method to imitate flatness-based controllers for mobile robots using neural networks. Sample case studies for a unicycle mobile robot and an unmanned aerial vehicle (UAV) quadcopter are presented. The goals of this paper are to (1) train a neural network to approximate a previously designed flatness-based controller, which takes in the desired trajectories previously planned in the flatness space and robot states in a general state space, and (2) present a dynamic training approach to learn models with high-dimensional inputs. It is shown that a simple feedforward neural network could adequately compute the highly nonlinear state variables transformation from general state space to flatness space and replace the complicated designed heuristic to avoid singularities in the control law. This paper also presents a new dynamic training method for models with high-dimensional independent inputs, serving as a reference for learning models with a multitude of inputs. Training procedures and simulations are presented to show both the effectiveness of this novel training approach and the performance of the well-trained neural network.

Low-frequency fiscal uncertainty

Long-run fiscal levels, or synonymously, fiscal targets, are usually assumed to be known to households inside the economy. This paper investigates the effects of unknown fiscal targets in an incomplete information, anticipated utility (IIAU) environment. Slowly increasing transfer payments cause households to suspect time-varying targets, and misperceptions impact both their expectation formation and decision making. Perceived targets enter the IIAU model as nonlinear state variables, introducing state-dependent fiscal multipliers. The short-run stimulus effects of Trump’s 2017 and Reagan’s 1981 Tax Cuts are comparable and modest. The long-run effects of Trump’s Tax Cuts worse off if transfers continue to increase.

Two Stage Particle Filter Based Terrain Referenced Navigation for Computational Efficiency

A novel particle filter (PF) based terrain referenced navigation (TRN) is presented for computational efficiency in this paper. TRN is a navigation method using terrain elevation information, mainly using a Bayesian filter such as a PF.PF based TRN has a disadvantage in that it requires a large number of particles to achieve a good estimation performance, which increases the computationalburden.To solve this disadvantage, two stage particle filter (TSPF) is proposed. TSPF consists of two stages and estimates the nonlinear state variables and the linear state variables separately. In the first stage, the nonlinear state variables are estimated by PF. Then, the linear state variables are estimated by a Kalman filter in the second stage considering a correlation between nonlinear and linear state variables. Since TSPF estimates only nonlinear state variables using PF and estimates linear state variables with only one Kalman filter, the computational efficiency can be greatly improved although the estimation performance is degraded a bit.In order to verify the proposed method, simulationsare performed to compare TSPF with PF andRao Blackwellized PF (RBPF)which has the same purpose as TSPF. The simulation results show similar estimation performance with fewer particles than with all state variables estimated using PF, resulting in less computational burden.

Monitoring of wastewater treatment plants using improved univariate statistical technique

Proper operation of the wastewater treatment plants (WWTPs) is crucial in order to maintain the sought effectiveness and desirable water quality. Therefore, the objective of this paper is to develop univariate statistical technique that aims at enhancing the monitoring of wastewater treatment plants using an improved particle filtering (IPF)-based multiscale optimized exponentially weighted moving average chart (MS-OEWMA). The advantages of the developed technique are fivefold: (i) estimate a nonlinear state variables of WWTPs using IPF technique. The IPF method yields an optimum choice of the sampling distribution, which also accounts for the observed data; (ii) use the dynamical multiscale representation to extract accurate deterministic features and decorrelate autocorrelated measurements. (iii) Develop an optimized EWMA (OEWMA) based on the best selection of smoothing parameter (λ) and control width L; (iv) combine the advantages of state estimation technique with MS-OEWMA chart to improve the fault detection in WWTP systems; and (v) investigate the effect of fault types (offset or bias, variance and drift) and fault sizes on the fault detection performances. The developed technique is validated using simulated COST wastewater treatment BSM1 model. The BSM1, provided by the IWA Task Group on Benchmarking of Control Strategies, is a simulation platform that allows for creating sensor faults disturbances in a wastewater treatment plant. The detection results are evaluated using three fault detection criteria: missed detection rate (MDR), false alarm rate (FAR) and average run length (ARL1).

Iterated square root unscented Kalman filter for nonlinear states and parameters estimation: three DOF damped system

Structural health monitoring of civil engineering infrastructure involves uncertainties for damage detection, damage identification, damage classification, sensor optimization, safety, durability, reliability, serviceability, performance based engineering, life-cycle performance, risk management, decision making and so on. For incorporating such uncertainties, several filtering techniques accounting for stochasticity can be implemented utilizing collected data from the structures. In this paper, an iterated square root unscented Kalman filter method is proposed for the estimation of the nonlinear state variables of nonlinear structural systems. Various conventional and state-of-the-art state estimation methods are compared for the estimation performance, namely the unscented Kalman filter (UKF), the square-root unscented Kalman filter (SRUKF), the iterated unscented Kalman filter (IUKF) and the iterated square root unscented Kalman filter (ISRUKF), in two comparative studies. In the first study, UKF, IUKF, SRUKF and ISRUKF methods are utilized at a simple non-linear second order LTI system with the aim to predict a two state variables, and to estimate two model parameters. In the second study, UKF, IUKF, SRUKF and ISRUKF techniques are utilized to a complex three degree of freedom spring-mass-dashpot system to predict the displacements and the velocities state variables. They are also used to estimate the model’s hysteretic parameters. Furthermore, the effect of practical challenges (e.g., measurement noise, number of states and parameters to be estimated) on the performances of UKF, IUKF, SRUKF and ISRUKF were investigated. The results of both comparative studies reveal that the ISRUKF method provides a better estimation accuracy than the IUKF method; while both methods provide improved accuracy over the UKF and SRUKF methods. The benefit of the ISRUKF method lies in its ability to provide accuracy related advantages over other estimation methods since it re-linearizes the measurement equation by iterating an approximate maximum a posteriori (MAP) estimate around the updated state, instead of relying on the predicted state. The results of the comparative studies show also that, for all the techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. The ISRUKF, however, still provides advantages over other methods in terms of the estimation accuracy and convergence.

Bond graph modeling for fault diagnosis: the continuous stirred tank reactor case study

Supervision of chemical and biochemical processes is difficult due to various factors, for example the evolution of nonlinear state variables, uncertainties in parameters and disturbances, and system operating conditions. The complexities arise out of the energetic couplings in the model. In this work, the bond graph representation, as a unified and multidisciplinary tool, is used for dynamic modeling and diagnosis for process engineering, incorporating the chemical kinetics of an equilibrated reaction, heat and mass transport phenomena. A pseudo bond graph model technique is applied to a pilot reactor fitted with a mono-fluid heating/cooling system. A modeling structure has been implemented through process sub-units and validated through experimental observations. This technique is proposed to indicate physico-chemical failures, for example matter leakage, caloric isolation, or the occurrence of a secondary reaction or temperature runaway when a very exothermic reaction occurs. It provides the computational algorithms to eliminate unknown variables from coupled thermochemical models and thus generate analytical redundancy relations (ARRs) in terms of measurements and parameters for online diagnosis.

Two-state dynamic friction model with elasto-plasticity

A novel two-state dynamic friction model with elasto-plasticity that we call 2SEP friction model is described. Being primarily intended for control, the modeled friction dynamics constitutes a linear combination of two nonlinear state variables. One independent and one dependent friction states capture the presliding hysteresis and transient sliding response. The overall dynamic friction with lag converges towards the velocity-dependent steady-state which is described by the Stribeck characteristic curve plus a linear viscous friction term. The proposed model requires seven parameters only of which already five are attributed to the steady-state friction characteristics. The 2SEP friction model is experimentally evaluated and compared with state-of-the-art friction models namely LuGre and GMS. The model identification is performed under real circumstances of the actuated motion close to the applications with electro-mechanical drives. The 2SEP friction model, being easy to implement and robust against real noisy signals, proves to be quite accurate in prediction and well favorably comparable with the LuGre and GMS one.

A sequential monte carlo filter for joint linear/nonlinear state estimation with application to DS-CDMA

A sequential Monte Carlo filter is considered which combines previously developed sequential importance sampling (SIS) techniques for conditional linear Gaussian models with measurement linearization for construction of approximate simulation densities. The resulting sequential Monte Carlo Kalman filter (SMC-KF) consists of a bank of conventional Kalman filters individually tuned to sampled trajectories of the nonlinear state variables. Sampling is according to a Gaussian distribution, with mean and covariance determined by extended Kalman filter-type equations. The SMC-KF is then applied to joint delay and multipath channel estimation in direct-sequence code-division multiple access (DS-CDMA). A combined analytical/simulation technique is employed to compare performance of the SMC-KF and a previously derived extended Kalman filter (EKF)-based DS-CDMA channel estimator.

Use of stiffness measure in nonlinear response analysis of concrete structures

A method used for measuring the stiffness of hinging reinforced concrete frame structures is developed. The structures suffer from a loss of stiffness during responses exceeding the yield deflection during earthquakes. The so-called stiffness measure method is used to evaluate the tangent stiffness of hinge regions while the structure is responding in nonlinear ranges. Eigenvector methods for nonlinear response have not been especially popular because of the need for regenerating eigenvectors as the time history proceeds. In the present work the eigenvector sets and corresponding nonlinear state variables, i.e. the tangent stiffnesses of the hinge regions, are stored. There is an expectation that previously generated eigenvectors can be reused as the analysis proceeds. The stiffness measure is used to compare the current tangent stiffnesses of hinge regions with those of previously stored eigenvectors sets. The stiffness measure quantity can be used to decide if a previously generated eigenvector set can be reused or if it is necessary to generate a new eigenvector set for the current time step. Since eigenvector calculations are diminished, the method is effective in reducing computational effort for reinforced concrete frame structures subjected to strong ground motions.