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System identification : theory for the user / Lennart Ljung
Abstract
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... To achieve this, it is necessary to optimize different areas, such as Electricity, Electronics, Mechanics, Administration, and Automatic Control, among others. In the area of Automatic Control, it is possible to contribute to the fulfillment of the optimization objective in the following ways: (1) tuning classical controllers [1]; (2) modeling and identification of parameters [2]; (3) monitoring and prediction of behaviors [3]; and (4) implementation of non-conventional controllers [4]. This work will focus on the modeling and identification of parameters, allowing a mathematical expression of the linear or nonlinear nature of production processes through the application of classical mechanics methods and computer science techniques [2]. ...
... To achieve this, it is necessary to optimize different areas, such as Electricity, Electronics, Mechanics, Administration, and Automatic Control, among others. In the area of Automatic Control, it is possible to contribute to the fulfillment of the optimization objective in the following ways: (1) tuning classical controllers [1]; (2) modeling and identification of parameters [2]; (3) monitoring and prediction of behaviors [3]; and (4) implementation of non-conventional controllers [4]. This work will focus on the modeling and identification of parameters, allowing a mathematical expression of the linear or nonlinear nature of production processes through the application of classical mechanics methods and computer science techniques [2]. ...
... In the area of Automatic Control, it is possible to contribute to the fulfillment of the optimization objective in the following ways: (1) tuning classical controllers [1]; (2) modeling and identification of parameters [2]; (3) monitoring and prediction of behaviors [3]; and (4) implementation of non-conventional controllers [4]. This work will focus on the modeling and identification of parameters, allowing a mathematical expression of the linear or nonlinear nature of production processes through the application of classical mechanics methods and computer science techniques [2]. As a possible benefit, it is possible to carry out simulations of the behavior and dynamics of a process under different considerations such as operating conditions, different controllers, and performance evaluation, among others. ...
This paper demonstrates that biodiesel production processes can be optimized through implementing a controller based on fuzzy logic and neural networks. The system dynamics are identified utilizing convolutional neural networks, enabling tests of the reactor temperature response under different control law proposals. In addition, a sensorless technique using a convolutional neural network to replace the sensor/transmitter signal in case of failure is implemented. Two optimization functions are proposed utilizing a metaheuristic algorithm based on differential evolution, where the aim is to minimize the use of cooling for the control of the reactor temperature. Finally, the control system proposals are compared, and the results show that a neuro-fuzzy controller without optimization restrictions generated unviable ITAE (1.9597×107) and TVU (22.3993) performance metrics, while the restriction proposed in this work managed to minimize these metrics, improving both the ITAE (3.3928×106) and TVU (17.9132). These results show that combining the sensorless technique and our optimization method for the cooling stage enables energy saving in the temperature control processes required for biodiesel production.
... For a good model, the cross-correlation function between residuals of the model and input variable does not go significantly outside the confidence region (Ljung, 1999). ...
... Then, it could be finding the following criteria (Ljung, 1999;Burnham & Anderson, 2002;Bisgaard & Kulahci, 2011;Jönsson, 2015;Brockwell & Davis, 2016): ...
... Equation 2.26 will be written as follows (Shumway & Stoffer, 2011;Brockwell & Davis, 2016): white noise as a discrete error (Ljung, 1999;Saaed, 2015; شعيث & جمعة ، 2017): ...
In this thesis, some linear dynamic systems, represented by Autoregressive with exogenous input variables (ARX models), where used to forecast crude oil prices (considered as dependent variable or output variable) for OPEC organization with the help of crude oil production (considered as independent variable or input variable) for the same organization depending on the data started from the period of 1973 until 2018.
First, the data were processed and analyzed through some statistical packages, such as MATLAB & SIMULINK R2018b, SPSS v25, EasyFit v5.5, and Microsoft Excel 2016. Using classical ARX method and proposed method (Bivariate Wavelet Filtering) for the time series data in order to select one of them for forecasting through comparing three measures of accuracy, MSE, FPE, and AIC. Then, applying crude oil prices for OPEC using the classical ARX models and ARX models with applying the bivariate wavelet filtering, especially bivariate Haar wavelet.
The main conclusions of the thesis were that the success of proposed model in forecasting of crude oil prices using bivariate wavelet filtering was more appropriate than classical models, ARX models have more forecasting parameter options than ARIMA models with the impact of input variable on output variable, and the forecasting of crude oil prices using proposed method in 2019 and 2020.
... The algorithm is a standard parameter estimation method. If outliers or noise are present in the historical data, existing methods for handling such issues will be employed [36]. For example, outliers will be treated as unknown parameters, while noise will be mitigated through data smoothing techniques (e.g., simple moving average) or noise filtering methods (e.g., wavelet denoising). ...
... (1) The mechanistic models for the desulfurization, steam boiler, air separation, and syngas compressor in (1)-(5) are first formulated, with model parameters to be estimated through the genetic algorithm. Next, the posterior PDFs of these parameters in (22) are fitted based on the Bayesian estimation theory, which are then employed to establish the multi-objective stochastic optimization model in (36). ...
... (2) One set of parameter samples, denoted as ξ SC3,1 , is randomly drawn from the posterior PDFs using the MCMC approach in Section 3.2.2. These samples serve as inputs for the optimization model in (36), transforming the uncertain optimization problem into a deterministic one. ...
In gas-to-methanol processes, optimizing multi-energy systems is a critical challenge toward efficient energy allocation. This paper proposes an entropy-based stochastic optimization method for a multi-energy system in a gas-to-methanol process, aiming to achieve optimal allocation of gas, steam, and electricity to ensure executability under modeling uncertainties. First, mechanistic models are developed for major chemical equipments, including the desulfurization, steam boilers, air separation, and syngas compressors. Structural errors in these models under varying operating conditions result in noticeable model uncertainties. Second, Bayesian estimation theory and the Markov Chain Monte Carlo approach are employed to analyze the differences between historical data and model predictions under varying operating conditions, thereby quantifying modeling uncertainties. Finally, subject to constraints in the model uncertainties, equipment capacities, and energy balance, a multi-objective stochastic optimization model is formulated to minimize gas loss, steam loss, and operating costs. The entropy weight approach is then applied to filter the Pareto front solution set, selecting a final optimal solution with minimal subjectivity and preferences. Case studies using Aspen Hysys-based simulations show that optimization solutions considering model uncertainties outperform the counterparts from a standard deterministic optimization in terms of executability.
... Ultimately, LDS models of sensorimotor learning are useful only if they can be fit to experimental data. The process of selecting the LDS model that best accounts for a sequence of inputs and outputs is called system identification (Ljung, 1999). Here we take a maximum-likelihood approach to system identification. ...
... Generally, identification of a system operating under closed loop (i.e., where the output is fed back to the learning rule) is more difficult than if the same system were operating in open loop (no feedback). This is partly because the closed loop makes the system less sensitive to external input (Ljung, 1999). In addition, and perhaps more important for our application, since the output directly affects the state, these two quantities tend to be correlated. ...
... This factorization means that for the purposes of this algorithm, we can regard the feedback as just another input variable. This view corresponds to the direct approach to closed-loop system identification (Ljung, 1999). The two steps of the EM algorithm for identifying the LDS model in equation 2.6, when B = D = 0 and C is known, were first reported by Shumway and Stoffer (1982). ...
Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation— the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a sub-stantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.
... The reason is that most adaptive control schemes include a model of the control object, which plays a crucial role in adjusting the control law. Obtaining such a model from data on the behavior of the control object constitutes the content of the classical dynamical system identification problem [31,32]. The well-known manual on system identification [31] says, "Inferring models from observations and studying their properties is really what science is about". ...
... Obtaining such a model from data on the behavior of the control object constitutes the content of the classical dynamical system identification problem [31,32]. The well-known manual on system identification [31] says, "Inferring models from observations and studying their properties is really what science is about". To a large part this is true, but still we should not forget that models are created for certain purposes, namely, to be able to analyze the behavior of modeled systems, and in the case of controlled systems we are interested in, also to provide the synthesis of control laws for them. ...
... In solving identification problems for controlled dynamical systems, a number of typical test excitations are used [31,[39][40][41]. Typical examples of such influences include the step action, rectangular pulse, and doublet (Figure 3), as well as more complex and informative variants such as the random signal ( Figure 4a) and the polyharmonic signal ( Figure 4b). ...
Motion control of modern and advanced aircraft has to be provided under conditions of incomplete and inaccurate knowledge of their parameters and characteristics, possible flight modes, and environmental influences. In addition, various abnormal situations may occur during flight, in particular, equipment failures and structural damage. These circumstances cause the problem of a rapid adjustment of the used control laws so that the control system can adapt to the mentioned changes. However, most adaptive control schemes have a model of the control object, which plays a crucial role in adjusting the control law. That is, it is required to solve also the identification problem for dynamical systems. We propose an approach to solving the above-mentioned problems based on artificial neural networks (ANNs) and hybrid technologies. In the class of traditional neural network technologies, we use recurrent neural networks of the NARX type, which allow us to obtain black-box models for controlled dynamical systems. It is shown that in a number of cases, in particular, for control objects with complicated dynamic properties, this approach turns out to be inefficient. One of the possible alternatives to this approach, investigated in the paper, consists of the transition to hybrid neural network models of the gray box type. These are semi-empirical models that combine in the resulting network structure both empirical data on the behavior of an object and theoretical knowledge about its nature. They allow solving with high accuracy the problems inaccessible by the level of complexity for ANN models of the black-box type. However, the process of forming such models requires a very large consumption of computational resources. For this reason, the paper considers another variant of the hybrid ANN model. In it, the hybrid model consists not of the combination of empirical and theoretical elements, resulting in a recurrent network of a special kind, but of the combination of elements of feedforward networks and recurrent networks. Such a variant opens up the possibility of involving deep learning technology in the construction of motion models for controlled systems. As a result of this study, data were obtained that allow us to evaluate the effectiveness of two variants of hybrid neural networks, which can be used to solve problems of modeling, identification, and control of aircraft. The capabilities and limitations of these variants are demonstrated on several examples. Namely, on the example of the problem of aircraft longitudinal angular motion, the possibilities of modeling the motion using the NARX network as applied to a supersonic transport aircraft (SST) are first considered. It is shown that under complicated operating conditions this network does not always provide acceptable modeling accuracy. Further, the same problem, but applied to a maneuverable aircraft, as a more complex object of modeling and identification, is solved using both a NARX network (black box) and a semi-empirical model (gray box). The significant advantage of the gray box model over the black box one is shown. The capabilities of the hybrid model realizing deep learning technologies are demonstrated by forming a model of the control object (SST) and neurocontroller on the example of the MRAC adaptive control scheme. The efficiency of the obtained solution is illustrated by comparing the response of the control object with a failure situation (a decrease in the efficiency of longitudinal control by 50%) with and without adaptation.
... Nowadays, methods of convex optimization play a crucial role in machine learning [1], system identification [2], optimal control [3,4], computational linear algebra [5,6], B Gerasim Krivovichev g.krivovichev@spbu.ru 1 Saint Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg 199034, Russian Federation and in other fields. Gradient methods of convex optimization are well-developed (e.g., see [7][8][9]), but in last decades new methods based on solving of the initial problem for the system of ordinary differential equations (ODEs) have been proposed [10][11][12][13][14][15][16][17][18][19][20][21][22]. ...
... So methods for obtaining a steady-state solution of (2) can be considered as methods for solving problem (1). As it is known [10,11], the explicit Euler method being applied to (2) leads to the standard GD. Application of other discretization methods for the solution of (2) will lead to new gradient methods, which can be considered as an alternative to well-known schemes. ...
... where ψ(h, λ) = 1 − hλ + (hλ) 2 2 − (hλ) 3 6 , ε > 0 is small enough, ...
The paper is devoted to the construction and analysis of gradient methods for convex optimization based on the explicit Goeken–Johnson’s Runge–Kutta methods for solving the Cauchy problem. Such two-step methods are based on the finite-difference approximation of solution derivatives and need fewer stages than standard Runge–Kutta methods. For quadratic problems, theorems on the convergence conditions and optimal stepsize for third- and fourth-order methods are presented. For the class of smooth convex functions, restricted by some assumptions, accelerated methods based on application to the Cauchy problem for non-autonomous system of second-order ordinary differential equations are constructed. The theorem on the convergence rate of accelerated methods is presented. As it is demonstrated for higher-order methods, the convergence rate can be better than for Nesterov’s accelerated gradient method, applied to functions from the considered class. Theoretical results are supported by numerical experiments for problems, which arise in different applications. Following problems are considered: minimization of convex quadratic and non-quadratic functions; minimization of integral functional; logistic regression problem; training of feedforward neural network. As it is demonstrated, proposed methods converge faster than the gradient descent method and Nesterov’s method. According to fewer number of stages, constructed methods require less computational time in comparison with algorithms based on standard explicit Runge–Kutta schemes.
... First, we show that the proposed algorithm outperforms the well-known offline subspace identification method N4SID [32] due to its capability of online adaptation. Second, we benchmark it against a standard online identification method, that estimates a parametric model via the recursive least squares technique [33]. While the two methods achieve similar prediction error nominally, the proposed gradient-based method exhibits better robustness properties against large measurement errors. ...
... We compare our approach with two methods from the system identification toolbox of MATLAB that are often used as benchmarks. Namely, we consider the N4SID algorithm that estimates a state-space model offline using subspace identification method [32], and with the recursiveLS algorithm, that estimates the parameters of a multivariate ARX model online using recursive least squares technique [33]. The identified models will then be used to predict future outputs, and the prediction error is compared. ...
... The noise covariance matrices for the filter are tuned on the validation data. For the ARX model, the output and input orders (c.f., [33]) are selected as n a = 1 and n b = 3, respectively, both from the interval {1, 2, . . . , 10}. ...
This paper introduces an online approach for identifying time-varying subspaces defined by linear dynamical systems, leveraging optimization on the Grassmannian manifold leading to the Grassmannian Recursive Algorithm for Tracking (GREAT) method. The approach of representing linear systems by non-parametric subspace models has received significant interest in the field of data-driven control recently. We view subspaces as points on the Grassmannian manifold, and therefore, tracking is achieved by performing optimization on the manifold. At each time step, a single measurement from the current subspace corrupted by a bounded error is available. The subspace estimate is updated online using Grassmannian gradient descent on a cost function incorporating a window of the most recent data. Under suitable assumptions on the signal-to-noise ratio of the online data and the subspace's rate of change, we establish theoretical guarantees for the resulting algorithm. More specifically, we prove an exponential convergence rate and provide a consistent uncertainty quantification of the estimates in terms of an upper bound on their distance to the true subspace. The applicability of the proposed algorithm is demonstrated by means of numerical examples, and it is shown to compare favorably with competing parametric system identification methods.
... is the observation array, that contains only the measurable signals. The advantage of the regression form in (18) is that the Θ(t) parameter matrix can be simply estimated using a Recursive Least Squares (RLS) Algorithm [61]. ...
... However, the calculation of (Ψ T (t)Ψ(t)) −1 is computationally demanding and it seems reasonable, especially in case of digital controllers, that instead of recalculatingΘ(t) in each control cycle, use an update law that utilizesΘ(t − δt) and the estimation error of q E (t) = ψ T (t)Θ(t − δt). Such method is called Recursive Least Square Algorithm and it is widely used in adaptive control and system identification [61,62]. The RLS method works under the assumption that P(t) = Ψ T (t)Ψ(t) is non singular and it has inverse for all t, so (21) can be rewritten as ...
... They are often applied when there is little prior information about the structure of the system under investigation and in cases where relatively simple models are incorporated into system control algorithms, usually for situations involving relatively small fluctuations about a required steady state. The resulting models may be quite limited in the range of their applicability but may form a useful basis for control system design [15], [16]. ...
... Classical methods of system identification and parameter estimation may be based on continuous-time descriptions involving ordinary or partial differential equations or on discrete-time models derived from the continuous-time equations (see, e.g. [15], [20]- [24]). For nonlinear lumped-parameter models having a structure derived from theory, parameter estimation methods are usually based on 'local' or 'global' optimisation techniques. ...
... Instead, the parameters are treated as adjustable factors used to optimize the model's fit to the observed data. (LJUNG, 1999). ...
... When quantifying the accuracy of a predictive model, the Mean Square Error (MSE) and Normalized Root Mean Square Error (NRMSE) are used to determine the fitness between the model and collected data (LJUNG, 1999). Besides that, the Aikaike's Final Prediction Error Criterion (FPE) were also used to measure these nonlinear models accuracy, for more details see Niedźwiecki and Cio lek (2017). ...
The primary aim of this study was to develop a comprehensive multi-variable model for Homogeneous Charge Compression Ignition (HCCI) engines and evaluate its efficacy by comparing it to simulated HCCI engine data. The complexity of the engine’s nonlinear dynamics necessitated the development of appropriate models to address the associated control challenges. Multiple Inputs Single Output (MISO) models were employed to capture the pressure and temperature variables. The estimated outputs were generated using Nonlinear Autoregressive with Exogenous Inputs (NARX) and Hammerstein-Wiener (HW) models, both were used as black box models. The performance of each model was assessed using metrics such as Normalized Root Mean Square Error (NRMSE), Mean Square Error (MSE), and Akaike’s Final Prediction Error (FPE). The results highlight the effectiveness of the Hammerstein-Wiener models in accurately representing the intricate dynamics of complex combustion, as observed in HCCI engines. The consistency of these models in delivering reliable outcomes further underscores their suitability for modeling such intricate systems.
... Based on our previous work [12], we implemented and tested the following models for the real-time prediction of future glycaemia and the generation of hypoglycaemic alerts: a run-to-run approach based on an autoregressive model with recursive parameter estimation (rAR) [15], which is an adaptive method based on a well-known recursive scheme [16]; an autoregressive integrated moving average (ARIMA) model [17], which was the best performer in our previous work [12]; and a feed forward neural network (NN) [18]. In this study, we also investigated the use of two deep learning models: a long-short term memory (LSTM) neural network and a convolutional neural network stacked to a LSTM (CNN-LSTM). ...
... A possible solution, partially investigated in [11], is the implementation of real-time filtering techniques as a pre-processing step, prior to model forecasting. However, these techniques introduce an additional delay in the prediction process [16], which can further reduce the time for preventive or corrective actions and thus diminish clinical benefits. ...
Background
Post bariatric hypoglycaemic (PBH) is a late complication of weight loss surgery, characterised by critically low blood glucose levels following meal-induced glycaemic excursions. The disabling consequences of PBH underline the need for the development of a decision support system (DSS) that can warn individuals about upcoming PBH events, thus enabling preventive actions to avoid impending episodes. In view of this, we developed various algorithms based on linear and deep learning models to forecast PBH episodes in the short-term.
Methods
We leveraged a dataset obtained from 50 patients with PBH after Roux-en-Y gastric bypass, monitored for up to 50 days under unrestricted real-life conditions. Algorithms’ performance was assessed by measuring Precision, Recall, F1-score, False-alarms-per-day and Time Gain (TG).
Results
The run-to-run forecasting algorithm based on recursive autoregressive model (rAR) outperformed the other techniques, achieving Precision of 64.38%, Recall of 84.43%, F1-score of 73.06%, a median TG of 10 min and 1 false alarm every 6 days. More complex deep learning models demonstrated similar median TG but inferior forecasting capabilities with F1-score ranging from 54.88% to 64.10%.
Conclusions
Real-time forecasting of PBH events using CGM data as a single input imposes high demands on various types of prediction algorithms, with CGM data noise and rapid postprandial glucose dynamics representing the key challenges. In this study, the run-to-run rAR yielded most satisfactory results with accurate PBH event predictive capacity and few false alarms, thereby indicating potential for the development of DSS for people with PBH.
... 1 System identification 2 Automatic control 3 Linear time-invariant system 4 Nonlinear behavior 5 Functional expansion 6 Block-oriented models 7 Black box 8 Volterra expansion 9 Wiener expansion 10 Functional 11 Process separability 12 Static nonlinearity 13 10 Kullback-Leibler divergence 11 Posterior probability distribution 12 Variational EM 13 Restricted Boltzmann machine (RBM) 14 Energy-based models (EBM) 15 Neural networks 16 log ( ) ≥ ∫ ( | ) log ( , ) ( | ) = (log ( , )( | ) ...
... ( 1: ,1: | 1: ) = ∏ ( |̃) ( | ) حالت فضای برای که حالتي مشابه مدل پارامترهای آموزش منظور به مي زير صورت به را مشاهدات پايين کران ديديم، غيرخطي نويسيم: موخر احتمال توزيع واقعي ساختار اساس بر تغييراتي، توزيع آن در که بود: خواهد زير صورت به مشاهدات پايين کران تعريف اين با با نزديک شباهت مشخصا مدل، اين آموزش برای مناسب هزينه تابع به نتيجتا که دارد تغييراتي رمزنگار خود آموزش مكانيزم و هزينه تابع بهينه قابل نزولي گراديان الگوريتم از استفاده با و سادگي است. سازی ترم تمامي اين در موجود شرطي توزيع های مدل گوسي توزيع يک از مي پيروی مي تعيين عصبي شبكه يک توسط آن پارامترهای که کند شود. ...
... System identification and multi-system modeling Learning from heterogeneous physical systems requires system identification, traditionally performed via parametric models (Ljung, 1999) or hybrid PDE-constrained approaches (Raissi et al., 2019). While Hamiltonian methods implicitly encode system parameters through energy landscapes, conventional HNNs often require training separate models per system. ...
Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.
... System identification involves developing mathematical models using experimental data, often incorporating insights from physical principles (Ljung, 1999). Data-driven parametric models of multi-input multi-output (MIMO) systems are essential to optimize performance of engineered systems as they enable the design of highperformance controllers and observers, provide design validation and feedback, and facilitate online monitoring and fault diagnosis (Steinbuch et al., 2022). ...
Multivariable parametric models are essential for optimizing the performance of high-tech systems. The main objective of this paper is to develop an identification strategy that provides accurate parametric models for complex multivariable systems. To achieve this, an additive model structure is adopted, offering advantages over traditional black-box model structures when considering physical systems. The introduced method minimizes a weighted least-squares criterion and uses an iterative linear regression algorithm to solve the estimation problem, achieving local optimality upon convergence. Experimental validation is conducted on a prototype wafer-stage system, featuring a large number of spatially distributed actuators and sensors and exhibiting complex flexible dynamic behavior, to evaluate performance and demonstrate the effectiveness of the proposed method.
... To parameterize R 0 and RC branches, prediction error minimization (PEM) method was used [27]. The PEM method is based on minimizing error (e(t)) between output measured data y(t) and output model data ŷ(t) , written as e(t) = y(t) −ŷ(t) . ...
Lithium-ion batteries are used in a wide range of applications today. With that in mind, it is crucial to create accurate models for simulations of cell behavior under various circumstances. In this work, a 2RC electrical battery model of lithium-ion battery was developed and was enhanced by a kinetic battery model for improved accuracy in dynamic behavior. By implementing the kinetic battery model into the 2RC electrical battery model, the root-mean-square-error was reduced by 3.9 mV in average in the driving profiles and on average by 10.1 mV in the dynamic discharge performance test. The developed models are designed and validated to operate in the temperature range −5 °C–45 °C.
Graphical abstract
... Learning dynamical models that explain the relation between input excitations and the corresponding observed output response of a physical system over time, a.k.a. "system identification" (SYSID), has been investigated since the 1950s [32], mostly for linear systems [19]. In particular, subspace identification methods like N4SID [25] and related methods [24] have been used with success in practice and are available in state-of-the-art software tools like the System Identification Toolbox for MATLAB [20] and the Python package SIPPY [4]. ...
In this paper, we propose a very efficient numerical method based on the L-BFGS-B algorithm for identifying linear and nonlinear discrete-time state-space models, possibly under and group-Lasso regularization for reducing model complexity. For the identification of linear models, we show that, compared to classical methods, the approach often provides better results, is much more general in terms of the loss and regularization terms used (such as penalties for enforcing system stability), and is also more stable from a numerical point of view. The proposed method not only enriches the existing set of linear system identification tools but can also be applied to identifying a very broad class of parametric nonlinear state-space models, including recurrent neural networks. We illustrate the approach on synthetic and experimental datasets and apply it to solve a challenging industrial robot benchmark for nonlinear multi-input/multi-output system identification. A Python implementation of the proposed identification method can be found in the package jax-sysid , accessible at https://github.com/bemporad/jax-sysid .
... In real-time applications, a background noise trend is leastsquares fitted, using an appropriate model (Ljung, 1987), up to time of the injection. At that point, coefficients in the leastsquares fit are fixed and are used to predict the noise trend during the time of injection. ...
Significant advances in the use of tiltmeter technology for hydraulic fracture mapping have been made during the past decade. Improvements have progressed in instrument design, site
preparation, array processing to reject noise, and models for interpretation of the signal. These improvements have increased the precision of determining the orientation, dimensions, and positions of hydraulic fractures; they have also allowed the use of this technology in a greater variety of formations, at different depths, and at different rates and volumes of fracture treatments.
The relatively complex response of coal to hydraulic fracture stimulation makes real-time monitoring and mapping of the fracturing necessary. Real-time mapping provides timely information for comparison of fracture design with actual execution and offers an opportunity to modify the stimulation process as new conditions are revealed. Complex fracturing has been observed in a wide variety of materials as expected at shallow depths. The cleats (joints) of coal can enhance the overall tendency for discontinuities to compound the complexity of fracture response to stress gradients at shallow depths. The coal-cleat system may lead to pathologies, such as fracture growth out-of-zone and growth of a fracture out of the original plane.
These induced fracture pathologies may deviate so far from actual design that dehydration of the slurry, leading to screenout and possible loss of the well, cannot be avoided unless a premonitory signal can be established in real-time data; such signals would allow steps to be taken to modify injection rates and slurry concentrations to mitigate risks. A case study of a coal-bed fracture mapping job completed in Alabama is used to illustrate
most of these points. In particular, a statistical test (F-test for the equality of variances) is used as a criterion for justifying an increase in source complexity when a single-fracture source does
not adequately describe the observed surface-deformation field.
... where humans (or any other controllers) are asked to randomly perturb the system without solving any tasks. In control theory terminology, the goal of this is to perform "system identification" with "sufficient excitation" [37]. We combine the two datasets D W := D W expert ∪ D W explore to train the world model. ...
We present generative predictive control (GPC), a learning control framework that (i) clones a generative diffusion-based policy from expert demonstrations, (ii) trains a predictive action-conditioned world model from both expert demonstrations and random explorations, and (iii) synthesizes an online planner that ranks and optimizes the action proposals from (i) by looking ahead into the future using the world model from (ii). Crucially, we show that conditional video diffusion allows learning (near) physics-accurate visual world models and enable robust visual foresight. Focusing on planar pushing with rich contact and collision, we show GPC dominates behavior cloning across state-based and vision-based, simulated and real-world experiments.
... While there is no definitive agreement on the originator of the Least Squares algorithm, it is commonly attributed to the notable contributions of Karl Friedrich Gauss in the late 18th century. This method involves minimizing the difference between estimated and actual values to reduce estimation errors, enabling the estimated linear parameters to closely align with those of the actual plant parameters (Ljung, 1987). ...
This article addresses the challenge of identifying linear systems for mathematical modeling of a quadcopter drone, with the purpose of utilizing them in predictive and stochastic control algorithms. The approach involves the application of two identification methods, namely the Ordinary Least Squares for ARX systems and the Extended Least Squares for ARMAX systems. The findings indicate that the ARMAX models exhibit superior performance indices and are deemed more appropriate for controller projects.
... No processo de atualização dos clusters tambémé necessário atualizar os consequentes e, nessa abordagem, issoé feito a partir do auxílio do algoritmo Mínimos Quadrados Recursivos -MQR Ljung (1999). Nesse procedimento, o algoritmo MQRé executado recursivamente em um ciclo p vezes, em que a cada ciclo atualiza-se somente a linha j da matriz π k i , para j = 1,2, · · · ,p. ...
The robust granular controller – CGR is a non-linear control method whose main characteristic is to add robustness to the Feedback Linearization technique. Despite demonstrating enough efficiency in what it proposes, the use of the CGR is subject to the choice of several parameters related to the learning process that influences the dynamics of the controller. This fact makes its application in real systems complex, especially for multivariable systems. This work proposes a methodology based on the differential evolution algorithm (DE) for tuning the CGR parameters. In addition, a reference model is used in the tuning process, making the closed-loop dynamic with uncertainties more similar to the desired dynamic for the nominal system. The proposed approach is applied to a level control system for non-linear tanks. The results indicate that the tuning method via DE provides significant performance improvement to the closed loop.
... The numeric FRF can either be obtained using additional, expensive measurement equipment such as an impact hammer and acceleration sensor [15,16] or be computed from the joint position and torque measurements. In the past 20 years frequency domain identification of elasticities of serial manipulators has been the subject of excessive research [17][18][19][20][21][22]. So far, none of these methods were applied to parallel robots. ...
Delta robots are prominent examples of agile parallel kinematic machines (PKM) designed for highly dynamic pick and place tasks. Optimized minimum time trajectories lead to dynamic load cycles, induce vibrations, and cause overshooting of the end-effector (EE) due to the flexibility of the PKM. Crucial to mitigate these effects by means of model-based control is a dynamics model that accounts for the principal elastic compliance, such as gear stiffness and structural elasticities. However, robot manufacturers do not provide data on the structural stiffness. Also established dynamics identification methods cannot determine stiffness and damping parameters. In this paper, a two-step frequency domain identification method is proposed to identify elastic properties by examples of an industrial Delta robot. As a peculiarity of the Delta PKM, the identification is carried out when the platform is removed and for the complete PKM. This allows distinguishing elasticities of the gear-drive units and of the struts. The identified parameters are employed for motion correction to avoid overshooting. This correction does not interfere with the original planning and control function of the industrial robot. Three motion correction schemes (preloading of drives, quasi static correction, flatness-based) are compared. Laser tracker measurements of the EE confirm a drastic reduction of overshooting and thus an increase of the overall tracking accuracy.
... Under closed-loop conditions, Eq. (15) becomes biased due to correlation between the input and output noise, which is caused by feedback. The bias can be avoided using a joint input-output estimate of the frequency response [23][24][25]40]. ...
Small unmanned aircraft systems face unique challenges in system identification due to their light weight, small size, and increased sensitivity to wind and gusts. To address these challenges, flight test procedures for sUAS system identification using open-loop and closed-loop flight data are presented. The approaches are demonstrated through system identification of a small flying-wing UAS equipped with a low-cost Pixhawk autopilot. The open-loop method is demonstrated through identification of the longitudinal bare-airframe aircraft dynamics. Longitudinal frequency responses were estimated from the open-loop flight data, and a linear state space model was identified from the estimated frequency responses and supplemental trim data, which was used to better identify parameters related to the low-frequency phugoid mode. The identified phugoid mode matches well with low-frequency oscillations measured from flight data in the time domain. The closed-loop method is demonstrated through identification of the lateral-directional dynamics. This approach is shown to improve signal-to-noise ratio and reduce deviation from the reference flight condition, which improves modeling results, and it can be used to simultaneously identify the bare-airframe, closed-loop, and broken-loop UAS dynamics from the same set of closed-loop flight data, reducing time and efforts spent flight testing. The identified lateral-directional model is compared with closed- and broken-loop frequency responses estimated from flight data. The results show excellent agreement, with all computed metrics, such as stability margins, matching within 10%, demonstrating the effectiveness of this method.
... A long research and development history has established robust techniques for analyzing system responses to sinusoidal excitation [61]. Accurately identifying frequency response characteristics based on sinusoidal excitation, combined with techniques like Fourier analysis, effectively filters out noise, providing a clearer picture of the system's true response. ...
Silicosis, the most dangerous and common lung illness associated with breathing in mineral dust, is a significant health concern. Spirometry, the traditional method for evaluating pulmonary functions, requires high patient compliance. Respiratory Oscillometry and electrical models are being studied to evaluate the respiratory system. This study aims to harness the power of machine learning (ML) to enhance the accuracy and interpretability of oscillometric parameters in silicosis. The data was obtained from 109 volunteers (60 in the training and 49 in the validation groups). Some supervised ML algorithms were selected for tests: K-Nearest Neighbors, Logistic Regression, Random Forest, CatBoost (CAT), Explainable Boosting Machines (EBM), and a deep learning algorithm. Two synthetic data generation algorithms were also applied. Initially, this study revealed the most accurate oscillometric parameter: the resonant frequency (fr, AUC=0.86), indicating a moderate accuracy (0.70-0.90). Next, original oscillometric parameters were used as input in the selected algorithms. EBM (AUC=0.93) and HyperTab (AUC=0.95) demonstrated the best performance. When feature selection was applied, HyperTab (AUC=0.94), EBM (AUC=0.94), and Catboost (AUC=0.93) emerged as the most accurate results. Finally, external validation resulted in a high diagnostic accuracy (AUC=0.96). Machine learning algorithms introduced enhanced accuracy in diagnosing respiratory changes associated with silicosis. The HyperTab and EBM achieved a high diagnostic accuracy range, and EBM explains the importance of the features and their interactions. This AI-assisted workflow has the potential to serve as a valuable decision support tool for clinicians, which can enhance their decision-making process, ultimately leading to improved accuracy and efficiency.
... Learning convolution kernels and convolution operators is close to the central identification of linear time-invariants in control theory and signal processing. We refer to [26] for a classic reference and to [5] for modern data-driven approaches that use optimization and machine learning. Estimating linear time-invariant systems involves learning the so-called impulse response or transfer function, which can often be represented as a convolution operator. ...
We consider the problem of learning convolution operators associated to compact Abelian groups. We study a regularization-based approach and provide corresponding learning guarantees, discussing natural regularity condition on the convolution kernel. More precisely, we assume the convolution kernel is a function in a translation invariant Hilbert space and analyze a natural ridge regression (RR) estimator. Building on existing results for RR, we characterize the accuracy of the estimator in terms of finite sample bounds. Interestingly, regularity assumptions which are classical in the analysis of RR, have a novel and natural interpretation in terms of space/frequency localization. Theoretical results are illustrated by numerical simulations.
... This approach is known as model or signal segmentation (Bassevill M. and Nikiforov I. (1993), Gustafsson F. (2001), Ljung L. (1999)) for which the output signal can be described from linear regression models with piecewise constant parameters: ...
The development of robust and reliable algorithm for modal parameters estimation made dynamic identification one of the most used approaches in the framework of structural health monitoring (SHM), while new technologies for sensors and data processing contributed to increase the accessibility and use of SHM methods in civil applications. Interest is growing with regard to the real-time SHM, which can support decision making in reducing the seismic risk in case of earthquakes. In this framework, the use of ARX models is proposed for automated off-line damage assessment of residential buildings subject to seismic events. To validate the algorithm, a numerical benchmark, in which input-output data are recorded by a virtual monitoring system that is trigged in case of earthquake, are considered.
... 2) ARX described by AR, is an Auto-regressive model that illustrates the associated noise to a signal X which is an exogenous input and is also called a Controlled Autoregressive model [22]. The relationship sets between the input and the output of the circuit characterizing the ARX model is given by a linear differential equation as Equation (6): ...
This present work aims to contribute to the solution of the problems encountered in electronic circuits fault diagnosis. One of these troubleshoots faced is the lack of effective features that help to optimize fault classifier and hence improve circuit fault detection and identification. Thus, our feature extraction approach is based on the CUT’s transfer function. This is deduced from the Matlab identification system IS model (ISM), namely the OE model belonging to the ARMA model’s family. These features are the transfer function polynomial coefficients playing a crucial role in the fault free and faulty circuits construction models and feeding the classifier for the fault diagnosis purpose. The faults we are dealing with are of single parametric type. This is done from PSPICE time domain analysis on the CUT output response under theses circuit conditions and followed by extracting the IS model (ISM) orders (p,q) polynomials. The coefficient values of the latter were considered as efficient comparison elements between faulty and healthy circuit responses. As a result, the OE model has achieved 100% fault coverage and its construction reached high accuracy level exceeding 98% for faulty circuits. This accuracy level ambition us to use its coefficients as input features for our Hybrid proposal fault classifier. This is built with GA and SVM algorithms combination targeting both data reduction and fault classification accuracy respectively. The results achieved are conclusive since the classifier accuracy level reached 100% and a 70% of feature data volume reduction was scored.
... The second component for forecast modeling is based on the relationship between hospitalizations and ILI and is defined in (7). This is an example of an autoregressive model with exogenous variables where the autoregressive lag is one (ARX (1)) (Raftery et al., 2010;Ljung, 1987). Figure 6 shows scatterplots of the difference between hospitalizations and scaled 1 week lags by ILI percentage at the US national level. ...
The annual influenza outbreak leads to significant public health and economic burdens making it desirable to have prompt and accurate probabilistic forecasts of the disease spread. The United States Centers for Disease Control and Prevention (CDC) hosts annually a national flu forecasting competition which has led to the development of a variety of flu forecast modeling methods. Beginning in 2013, the target to be forecast was weekly percentage of patients with an influenza-like illness (ILI), but in 2021 the target was changed to weekly hospitalizations. Reliable hospitalization data has only been available since 2021, but ILI data has been available since 2010 and has been successfully forecast for several seasons. In this manuscript, we introduce a two component modeling framework for forecasting hospitalizations utilizing both hospitalization and ILI data. The first component is for modeling ILI data using a nonlinear Bayesian model. The second component is for modeling hospitalizations as a function of ILI. For hospitalization forecasts, ILI is first forecast then hospitalizations are forecast with ILI forecasts used as a predictor. In a simulation study, the hospitalization forecast model is assessed and two previously successful ILI forecast models are compared. Also assessed is the usefulness of including a systematic model discrepancy term in the ILI model. Forecasts of state and national hospitalizations for the 2023-24 flu season are made, and different modeling decisions are compared. We found that including a discrepancy component in the ILI model tends to improve forecasts during certain weeks of the year. We also found that other modeling decisions such as the exact nonlinear function to be used in the ILI model or the error distribution for hospitalization models may or may not be better than other decisions, depending on the season, location, or week of the forecast.
... More specifically, an autoregressiveexogenous (ARX) model structure was selected. This method was selected because of its relatively low model complexity, while still being able to capture important system dynamics (Ljung, 1987). In comparison to complex and non-linear models, (e.g., Stirling et al., 2008;Zakynthinaki, 2015), the main advantage is that insightful model features such as TC and (steady state) gain values can be easily obtained from the linear model structure. ...
With the development of power output sensors in the field of paddle sports and the ongoing advancements in dynamical analysis of exercise data, this study aims to model the measurements of external training intensity in relation to heart rate (HR) time‐series during flat‐water kayak sprint. Nine elite athletes performed a total of 47 interval training sessions with incremental intensity (light to (sub‐) maximal effort levels). The data of HR, speed and power output were measured continuously and rating of perceived exertion and blood lactate concentration ([BLa]) were sampled at the end of each interval stage. Different autoregressive‐exogenous (ARX) modelling configurations are tested, and we report on which combination of input (speed or power), model order (1st or 2nd), parameter estimation method (time‐(in)variant) and training conditions (ergometer or on‐water) is best suited for linking external to internal measures. Average model R² values varied between 0.60 and 0.97, with corresponding average root mean square error values of 15.6 and 3.2 bpm. 1st order models with time‐varying (TV) parameter estimates yield the best model performance (average R² = 0.94). At the level of the individual athlete, the TV modelling features (i.e., the model parameters and derivatives such as time constant values) show significant repeated measure correlations in relation to measures of exercise intensity. In conclusion, the study provides a comprehensive description of how the dynamic relationship between external load and HR for sprint kayaking training data can be modelled. Such models can be used as a basis for improving training evaluation and optimisation.
Many real-world multivariable systems need to be modeled to capture the interconnected behavior of their physical variables and to understand how uncertainty in actuators and sensors affects the system dynamics. In system identification, some estimation algorithms are formulated as multivariate data problems by assuming symmetric noise distributions, yielding deterministic system models. Nevertheless, modern multivariable systems must incorporate the uncertainty behavior as a part of the system model structure, taking advantage of asymmetric distributions to model the uncertainty. This paper addresses the uncertainty modeling and identification of a class of multivariable linear dynamic systems, adopting a Stochastic Embedding approach. We consider a nominal system model and a Gaussian mixture distributed error-model driven by an exogenous input signal. The error-model parameters are treated as latent variables and a Maximum Likelihood algorithm that functions by marginalizing the latent variables is obtained. An Expectation-Maximization algorithm that jointly uses the measurements from multiple independent experiments is developed, yielding closed-form expressions for the Gaussian mixture estimators and the noise variance. Numerical simulations demonstrate that our approach yields accurate estimates of both the multivariable nominal system model parameters and the noise variance, even when the error-model non-Gaussian distribution does not correspond to a Gaussian mixture model.
We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.
The application of the electronic control unit (ECU) motivates dynamic models with high precision to simulate mechatronic systems for various analysis and design tasks like hardware-in-the-loop (HiL) simulation. Unlike traditional physical models which are established based on the research experience or the mechanics mechanism, in this study, a novel data-driven modeling approach is presented based on the piecewise affine (PWA) identification method. In this work, the highly nonlinear dynamic of automotive actuators is well approximated by the PWA model. To obtain experimental data that can accurately reflect the characteristics of actuators, a test bench was first built. On this basis, the PWA identification of automotive is composed of the data clustering, the model structure selection, and the model parameter estimation. The proposed clustering method improves the widely used balanced iterative reducing and clustering using hierarchies (BIRCH) by introducing a refinement phase for handling clusters with arbitrary shapes. The model structure selection and the parameter estimation are jointly solved by using the optimization method. The presented method is demonstrated with an academic example and an automotive throttle, and the results show that the proposed method can achieve a high model quality, which means that the normalized root mean squared error (NRMSE) is 0.03 and the absolute maximal prediction error can reach 2.43°. Compared to other models like a physical model or a fuzzy model, the improvement using the proposed model can be up to 50%. Thus, the quality of the proposed PWA model is sufficient for HiL simulation.
A new method to discover open-loop, unstable, longitudinal aerodynamic parameters, using a ‘two-stage optimization approach’ for designing optimal inputs, and with an application on the fighter aircraft platform, has been presented. System identification of supersonic aircraft requires formulating optimal inputs due to the extremely limited maneuver time, high angles of attack, restricted flight conditions, and the demand for an enhanced computational effect. A pre-requisite of the parametric model identification is to have a priori aerodynamic parameter estimates, which were acquired using linear regression and Least Squares (LS) estimation, based upon simulated time histories of outputs from heuristic inputs, using an F-16 Flight Dynamic Model (FDM). In the ‘first stage’, discrete-time pseudo-random binary signal (PRBS) inputs were optimized using a minimization algorithm, in accordance with aircraft spectral features and aerodynamic constraints. In the ‘second stage’, an innovative concept of integrating the Fisher Informative Matrix with cost function based upon D-optimality criteria and Crest Factor has been utilized to further optimize the PRBS parameters, such as its frequency, amplitude, order, and periodicity. This unique optimum design also solves the problem of non-convexity, model over-parameterization, and misspecification; these are usually caused by the use of traditional heuristic (doublets and multistep) optimal inputs. After completing the optimal input framework, parameter estimation was performed using Maximum Likelihood Estimation. A performance comparison of four different PRBS inputs was made as part of our investigations. The model performance was validated by using statistical metrics, namely the following: residual analysis, standard errors, t statistics, fit error, and coefficient of determination (R2). Results have shown promising model predictions, with an accuracy of more than 95%, by using a Single Sequence Band-limited PRBS optimum input. This research concludes that, for the identification of the decoupled longitudinal Linear Time Invariant (LTI) aerodynamic model of supersonic aircraft, optimum PRBS shows better results than the traditional frequency sweeps, such as multi-sine, doublets, square waves, and impulse inputs. This work also provides the ability to corroborate control and stability derivatives obtained from Computational Fluid Dynamics (CFD) and wind tunnel testing. This further refines control law design, dynamic analysis, flying qualities assessments, accident investigations, and the subsequent design of an effective ground-based training simulator.
Despite significant research, discovering causal relationships from fMRI remains a challenge. Popular methods such as Granger causality and dynamic causal modeling fall short in handling contemporaneous effects and latent common causes. Methods from causal structure learning literature can address these limitations but often scale poorly with network size and need acyclicity. In this study, we first provide a taxonomy of existing methods and compare their accuracy and efficiency on simulated fMRI from simple topologies. This analysis demonstrates a pressing need for more accurate and scalable methods, motivating the design of Causal discovery for Large-scale Low-resolution Time-series with Feedback (CaLLTiF). CaLLTiF is a constraint-based method that uses conditional independence between contemporaneous and lagged variables to extract causal relationships. On simulated fMRI from the macaque connectome, CaLLTiF achieves significantly higher accuracy and scalability than all tested alternatives. From resting-state human fMRI, CaLLTiF learns causal connectomes that are highly consistent across individuals, show clear top-down flow of causal effect from attention and default mode to sensorimotor networks, exhibit Euclidean distance dependence in causal interactions, and are highly dominated by contemporaneous effects. Overall, this work takes a major step in enhancing causal discovery from whole-brain fMRI and defines a new standard for future investigations.
We discuss determining a finite sample linear time-invariant (FS-LTI) system’s impulse response function, h[n], in the frequency domain when the input testing function is a uniform window function with a width of L and the output is limited to a finite number of effective samples, M. Assuming that the samples beyond M are all zeros, the corresponding infinite sample LTI (IS-LTI) system is a marginally stable system. The ratio of the discrete Fourier transforms (DFT) of the output to input of such an FS-LTI system, H0[k], cannot be directly used to find h[n] via inverse DFT (IDFT). Nevertheless, H0[k] contains sufficient information to determine the system’s impulse response function (IRF). In the frequency-domain approach, we zero-pad the output array to a length of N. We present methods to recover h[n] from H0[k] for two scenarios: (1) N≥max(L,M+1) and N is a coprime of L, and (2) N≥L+M+1. The marginal stable system discussed here is an artifact due to the zero-value assumption on unavailable samples. The IRF obtained applies to any LTI system up to the number of effective data samples, M. In demonstrating the equivalence of H0[k] and h[n], we derive two interesting DFT pairs. These DFT pairs can be used to find trigonometric sums that are otherwise difficult to prove. The frequency-domain approach makes mitigating the effects of interferences and random noise easier. In an example application in radar remote sensing, we show that the frequency-domain processing method can be used to obtain finer details than the range resolution provided by the radar system’s transmitter.
Establishing a suitable model of the studied nonlinear dynamic system is the basis and prerequisite for system analysis and design. Radial basis functions have the characteristics of simple form and flexible node configuration, which make it possible to form network models to fit complex nonlinear properties. Unlike most radial basis function network model estimation techniques assumed known time-delay, this paper concentrates on the combined parameter and time-delay estimation for this type of network models. To deal with the unknown time-delay, some additional variables are incorporated to formulate an extended identification framework grounded in redundant rule. Building upon this framework, a rolling window hierarchical gradient recursive sub-algorithm is derived to compute the parameter estimates using the recombined observation technique, and a threshold strategy is presented to filter out the redundant parameter estimates when determining the time-delay. Subsequently, a joint parameter and time-delay estimation algorithm is proposed to identify nonlinear dynamic networks with time-delays. The convergence property of the algorithm is analyzed, and its performance is validated through two case studies.
In self-adaptive software systems, the role of context is paramount, especially for proactive self-adaptation. Current research, however, does not fully explore context’s impact, for example on priorities of the requirements. To address this gap, we introduce a novel contextual goal model to capture these factors and their influence on the system. Using this, we propose a two-layer control mechanism with a context-aware model predictive control to achieve proactive adaptation for the software system and adaptation for the controller itself. By contextual prediction and a more accurate system model, our approach utilizes model predictive control to facilitate timely and efficient system adaptations, improving both performance and adaptability. Meanwhile, we perform requirement adaptation to update the contextual goal model, which in turn updates the objective function and constraints of the controller. Our experimental evaluations across two scenarios demonstrate the significant benefits of our approach in enhancing system performance.
In this work, full order nonlinear model of Wind Turbine Doubly-Fed Induction Generator (DFIG-WT), including
stator dynamics has been considered. Nonlinear Multi-Input-Multi-Mutput (MIMO) feedback linearization
controller has been designed to control the rotor speed of Doubly-Fed Induction Generator (DFIG) in Wind
Energy Conversion System (WECS). The desired states are chosen to drive the system at optimum rotor speed for
maximum power point tracking. Due to unpredictable nature of wind speed, the rotor may fail to retain the optimal
speed. Model Predictive Controller (MPC) with output constrain has been designed for the rotor side converter to
ensure that the rotor retain the optimal speed for all operating points. It also improves the transient state of the
rotor speed.
In this work, full order nonlinear model of Wind Turbine Doubly-Fed Induction Generator (DFIG-WT), including
stator dynamics has been considered. Nonlinear Multi-Input-Multi-Mutput (MIMO) feedback linearization
controller has been designed to control the rotor speed of Doubly-Fed Induction Generator (DFIG) in Wind
Energy Conversion System (WECS). The desired states are chosen to drive the system at optimum rotor speed for
maximum power point tracking. Due to unpredictable nature of wind speed, the rotor may fail to retain the optimal
speed. Model Predictive Controller (MPC) with output constrain has been designed for the rotor side converter to
ensure that the rotor retain the optimal speed for all operating points. It also improves the transient state of the
rotor speed.
In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold . We provide explicit bounds on in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.
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