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System Identification: Theory For The User
Abstract
The sections in this article are1The Problem2Background and Literature3Outline4Displaying the Basic Ideas: Arx Models and the Linear Least Squares Method5Model Structures I: Linear Models6Model Structures Ii: Nonlinear Black-Box Models7General Parameter Estimation Techniques8Special Estimation Techniques for Linear Black-Box Models9Data Quality10Model Validation and Model Selection11Back to Data: The Practical Side of Identification
... For a constant weighting function, the Bayes estimator is equivalent to the maximum likelihood (ML) estimator. Its consistency and asymptotic normality have been discussed under a variety of conditions, e.g., [5], [8], [13], [14], [20], [25]. For a general given weighting function, the asymptotic equivalence between the Bayes and ML estimators can be established using the Bernstein-von Mises Theorem [44, Chapter 10.2]. ...
... In this paper, we consider linear regression models and EB estimators with a quadratic loss function. Under certain conditions, the ML and EB estimators can both be proved to be consistent and asymptotically efficient [18], [25]. This means that they have the same first-order asymptotic statistical properties. ...
... Its high-order asymptotic properties depend on the asymptotic properties of its hyperparameter estimators. The consistency of a general hyperparameter estimator has been derived in [30,Section B.2] by using [25,Theorem 8.2]. More specifically, the consistency of the EB, the Stein's unbiased risk estimation (SURE), and the generalized cross-validation (GCV) hyper-parameter estimators have been established in [28], [30], [35]. ...
Empirical Bayes estimators are based on minimizing the average risk with the hyper-parameters in the weighting function being estimated from observed data. The performance of an empirical Bayes estimator is typically evaluated by its mean squared error (MSE). However, the explicit expression for its MSE is generally unavailable for finite sample sizes. To address this issue, we define a high-order analytical criterion: the excess MSE. It quantifies the performance difference between the maximum likelihood and empirical Bayes estimators. An explicit expression for the excess MSE of an empirical Bayes estimator employing a general data-dependent hyper-parameter estimator is derived. As specific instances, we provide excess MSE expressions for kernel-based regularized estimators using the scaled empirical Bayes, Stein unbiased risk estimation, and generalized cross-validation hyper-parameter estimators. Moreover, we propose a modification to the excess MSE expressions for regularized estimators for moderate sample sizes and show its improvement on accuracy in numerical simulations.
... Hence, it is reasonable to assume that the true system is of this class. In addition,are also several algorithms which return minimal systems in innovation form, e.g., subspace identification methods [33], [15] and some parametric methods [18]. For the latter, the use of minimal systems in innovation form also justifies viewing LTI systems as optimal predictors of the current output based on past outputs and inputs. ...
... 2. Run Algorithm 1 with M = Ψ u,y and selection (ᾱ,β) and denote the result by (26), (27) and (26) and (27). LetK I σ ,Q I σ as in (18). ...
... What remains is to prove that the latter sLSS is minimal. Note, from equations (16), (18) and (19), that the associated dLSSŜŜŜ and the minimal dLSSŜ are the same when N → ∞. It follows thatŜŜ is minimal because his associated dLSS is minimal. ...
... It helps optimize performance, enhances process control, and enables predictive capabilities. This step is vital for translating experimental data into actionable insights that inform the design and operation of dynamic processes [3]. By fitting models to dynamic data, experimentalists can convert complex, time-dependent information into predictive tools that guide decision-making, improve process safety, and drive innovation in fields such as chemical engineering, biotechnology, and environmental engineering. ...
... An additional necessary assumption of this formalism is that the underlying structure of the physical system can be uniquely identified from an appropriate set of measurements of both its input and output variables. Practical conditions to establish this property have been considered in the literature [3]. Although these tasks serve distinct purposes, they are addressed together in Section 3, as they are frequently performed concurrently. ...
... This task involves ensuring dimensional consistency across all terms in the model equations [3]. To achieve this, we need to (i) verify that units are clearly defined and that each equation maintains dimensional balance; (ii) assess the model's plausibility and its adherence to fundamental principles, particularly conservation laws; and (iii) ensure that initial and/or boundary conditions are consistent and provided. ...
Model fitting of laboratory-generated experimental data is a foundational task in engineering, bridging theoretical models with real-world data to enhance predictive accuracy. This process is particularly valuable in batch dynamic experiments, where mechanistic models are often used to represent complex system behavior. Here, we propose a systematic algorithm tailored for the model fitting and parameter estimation of experimental data from batch laboratory experiments, rooted in a Process Systems Engineering framework. The paper provides an in-depth, step-by-step approach covering data collection, model selection, parameter estimation, and accuracy assessment, offering clear guidelines for experimentalists. To demonstrate the algorithm’s effectiveness, we apply it to a series of dynamic experiments on the pressure-constant cake filtration of calcium carbonate, where the pressure drop across the filter is varied as a key experimental factor. This example underscores the algorithm’s utility in enhancing the reliability and interpretability of model-based analyses in engineering.
... Definition 1 [21]. The parameter q P is structurally globally identifiable if for almost any ...
... Definition 2 [21]. Let there be a neighbourhood ...
... If the neighbourhood of () P V does not exist, then the parameter P is called structurally locally unidentifiable. Definition 3 [21]. The model ...
The identifiability problem nonlinear decentralised systems (NDS) were not considered. Synthesis of mathematical models for NDS having a complex structure is an urgent problem. The presence of internal relationships and nonlinearities in NDS complicates the identifiability estimation. Therefore, models with uncertainty are used to design control systems for NDS. To describe the processes in NDS, it is necessary to apply adequate models based on verifying their identifiability. We propose an approach for assessing the local structural identifiability of nonlinear
decentralised systems. Conditions of local parametric identifiability are obtained as functional constraints for parameters and nonlinearity.The effect of the constant excitation of experimental data on the S-synchronizability of DS is shown. The S-synchronizability condition fulfilment guarantees the structural identifiability estimate of DS with nonlinearity belonging to a certain sector. We introduce the S-congruence concept and propose a method for estimating the S-synchronizability of an input. Geometric structures and criteria are used to assess NDS identifiability. The simulation
results are presented.
... ARX model needs both the input (measured cutting forces) and the output (cutting sound pressure measured with a microphone) to predict the mode frequencies of in-process thin-walled parts. Hence, an ARX model, in general, describes the system's output as a linear function of its past outputs and inputs as [34]: ...
... Coefficients of the AR and ARX models are obtained using the least squares method by representing Eq. (1) in vector form [34]: ...
Prediction of material removal-induced variations in the in-process thin-walled workpiece dynamics is crucial to develop a vibration-free machining process. Therefore, monitoring the varying vibration modes to adjust the cutting parameters along the toolpath is imperative. Natural frequencies are the key dynamic parameters in machining for monitoring and controlling the machining-induced vibrations and preventing resonance and machining instability. This study aims to identify the natural frequencies of thin-walled parts during milling using a non-contact sensor. An output-only autoregressive (AR) model with cutting sound signals and an autoregressive with exogenous inputs (ARX) model using the measured cutting forces and cutting sound are developed to predict the in-process dominant mode frequencies of a flexible part as the material is removed. The natural frequencies obtained by the developed models are compared with the experimental modal analysis results. The comparisons showed that the developed AR and ARX models can respectively predict the dominant in-process workpiece mode frequencies with maximum errors of 8.6% and 7.8%. These results reveal the potential of the proposed methodology for monitoring the machining of thin-walled parts to improve part quality and process productivity.
... 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 ...
Robust Fixed Point Transformation-based adaptive control design was introduced in 2009, in order to overcome some of the difficulties of certain Lypunov function-based adaptive control design techniques. This control method was developed for trajectory tracking applications for nonlinear second order systems. The RFPT controller, instead of refining some kind of available dynamic model, that is neiter precise nor complete, targets the deterministic behavior of the trajectory tracking error, by finding the correct control signal in an iterative manner. In each control cycle a single step of iteration is made. In this paper a variation of the RFPT control method is introduced, where concurrent adaptive control and model identification is applied, based on Recursive Least Square Algorithm and a Hammerstein model. In the proposed control scheme the estimated control response is used to apply multiple iterative steps, that way increasing the adaptation speed of the controller. The proposed solution was tested on experimental basis in a DC motor control application. The increased adaptation speed of the proposed method resulted in more precise trajectory tracking.
... Non-parametric system identification techniques, such as Frequency Response Functions (FRFs) [7], model systems without requiring a predefined structure. Unlike non-parametric approaches, parametric system identification generally involves a fixed model structure, e.g., using prediction error methods [8,9]. A key development in system identification involves the application of kernel methods [10]. ...
... 2. Apply input u h to the system and record y l . 3. Construct the regressor matrix Φ according to (9). 4. (Optional:) Tune the hyperparameters η and γ, e.g., based on marginal likelihood optimization (26). 5. Construct kernel matrix K using (19). ...
Models that contain intersample behavior are important for control design of systems with slow-rate outputs. The aim of this paper is to develop a system identification technique for fast-rate models of systems where only slow-rate output measurements are available, e.g., vision-in-the-loop systems. In this paper, the intersample response is estimated by identifying fast-rate models through least-squares criteria, and the limitations of these models are determined. In addition, a method is developed that surpasses these limitations and is capable of estimating unique fast-rate models of arbitrary order by regularizing the least-squares estimate. The developed method utilizes fast-rate inputs and slow-rate outputs and identifies fast-rate models accurately in a single identification experiment. Finally, both simulation and experimental validation on a prototype wafer stage demonstrate the effectiveness of the framework.
... The system identification problem has been extensively studied in the literature [1], [2]. The least-squares estimator (LSE) is the most widely studied approach for this problem as it provides a closed-form solution. ...
This paper investigates a subgradient-based algorithm to solve the system identification problem for linear time-invariant systems with non-smooth objectives. This is essential for robust system identification in safety-critical applications. While existing work provides theoretical exact recovery guarantees using optimization solvers, the design of fast learning algorithms with convergence guarantees for practical use remains unexplored. We analyze the subgradient method in this setting where the optimization problems to be solved change over time as new measurements are taken, and we establish linear convergence results for both the best and Polyak step sizes after a burn-in period. Additionally, we characterize the asymptotic convergence of the best average sub-optimality gap under diminishing and constant step sizes. Finally, we compare the time complexity of standard solvers with the subgradient algorithm and support our findings with experimental results. This is the first work to analyze subgradient algorithms for system identification with non-smooth objectives.
... In well-known and highly cited bibliography devoted to the identification of time-series models (see e.g. [4,9,20,24,26,28,30,32]), existence of this nonlinearity is omitted. An extensive literature search concerning the identification of time-series models based on quantized measurements resulted in only three items of literature [1,2,14] discussing the identification of only AR time-series models. ...
In the paper, two algorithms that allow identification of a parametric models of random time-series from binary-valued observations of their realizations, as well as from quantized measurements of their values, are proposed. The proposed algorithms are based on the idea of time-series decomposition either on a direct power spectral density or autocorrelation function approximation. They use the concepts of randomized search algorithms to recover the corresponding parametric models from calculated estimates of power spectral density or autocorrelation function. The considerations presented in the paper are illustrated with simulated identification examples in which linear and nonlinear block-oriented dynamic models of timeseries are identified from the binary-valued observations and quantized measurements.
... Although the term system identification was introduced by Lotfi Zadeh in the late 1950s [2], some progress (such as continuous functional [3], analytic functional [4]) was made by Maurice Fréchet and Vito Volterra before then. Since then, system identification has become an established research area within automatic control [5]. Considerable research has been conducted on this. ...
Control engineering plays an indispensable role in enhancing safety, improving comfort, and reducing fuel consumption and emissions for various industries, for which system identification, control, and optimization are primary topics. Alternatively, artificial intelligence (AI) is a leading, multi-disciplinary technology, which tries to incorporate human learning and reasoning into machines or systems. AI exploits data to improve accuracy, efficiency, and intelligence, which is beneficial, especially in complex and challenging cases. The rapid progress of AI facilitates major changes in control engineering and is helping advance the next generation of system identification, control, and optimization methods. In this study, we review the developments, key technologies, and recent advancements of AI-based system identification, control, and optimization methods, as well as present potential future research directions.
... where ∈ ℝ is the regression matrix and ∈ ℝ is the vector of samples. The parameter vector will be obtained in terms of the least squares method analytically as [52] . ...
Background/Objectives: The effectiveness of chemotherapy in cancer treatment is often compromised by inter-patient variability, leading to suboptimal outcomes. Traditional dosing protocols rely on population-based models that do not account for individual patient responses and the cancer phenotype. This study aims to develop a personalized chemotherapy dosing strategy by estimating uncertain model parameters using artificial neural networks, ensuring an optimal and individualized treatment approach. Methods: A dynamical model of tumor growth, immune response, and chemotherapy effects is used as the foundation for personalization. A training dataset is generated by simulating state responses across a diverse population of virtual patients, capturing inter-subject variability. The state responses are parameterized (approximated) using the sum of exponential functions to reduce dimensionality, and a multilayer perceptron artificial neural network is trained to estimate patient-specific model parameters based on response data from a single chemotherapy dose. Results: The proposed method effectively estimates patient-specific model parameters, significantly reducing uncertainty compared to conventional population-based models or the nonlinear least squares method. Numerical experiments demonstrate that personalized chemotherapy dosing, optimized using the estimated model parameters, achieves fast tumor remission while minimizing the total drug amount. Conclusions: By integrating the artificial neural network as the parameter estimator with model-based optimization, this study presents a novel approach to personalized chemotherapy dosing.
... 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 (sUAS) face unique challenges in system identification due to their lightweight, 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 the 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.
... The problem of stochastic system identification has been investigated for several decades. The earliest methods mainly address either discrete-time models [25,35] or continuous-time models that are however linear in the state variable [20], which thus do not fit the identification setting introduced by (1.1). ...
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.
... which follows from the Cramér-Rao inequality [26,19] and the fact that the error covariance matrix will be no worse than C t . The weighted norm in (7) aligns the future errors to the axes of the current confidence ellipsoid. ...
We consider the problem of joint input design and parameter estimation for identifying nonlinear system models through the sequential acquisition of measurements while adhering to system constraints. We utilize a receding horizon approach and propose a new scale-invariant input design criterion, which is tailored to continuously updated parameter estimates, along with a new sequential parameter estimator. We demonstrate the ability of the method to design informative experiments online, while steering the system within operational constraints.
... As a result, the Identification of models and controller design comprise the two stages of the control issue. This two-stage controller design technique, also known as system identification, involves synthesizing data from the system and then utilizing various mathematical methodologies to derive a mathematical model that optimally describes the system ( [1,2,3,4]). After obtaining an accurate model, a model-based control method is employed to select an appropriate controller for the setup. ...
This paper examines the switch from model-based to data-driven control systems, providing an overview of the challenges encountered and the potential solutions. It reviews the history, current progress, and future perspectives of data-driven control technologies. Additionally, it outlines the differences between model-based and data-driven control, compares various data-driven control approaches, and addresses essential issues in data-driven optimization. Finally, the paper delves into the data-driven control process and related research areas.
... RL algorithms include "model-free" approaches such as Q-learning, which learn the "quality" of a particular action executed while the system is in a particular state [39,55,75]. "Model-based" approaches, on the other hand, learn a model for the system [63,68] and are related to "system identification" from the control field [49,63]. RL has been used in applications such as board games (Go, chess), arcade games (PAC-MAN) [37,65], recommender systems [19,84], transport scheduling [10,18,45,66], finance [1,43], and autonomous driving [29,35,39,42]. ...
Detecting phishing, spam, fake accounts, data scraping, and other malicious activity in online social networks (OSNs) is a problem that has been studied for well over a decade, with a number of important results. Nearly all existing works on abuse detection have as their goal producing the best possible binary classifier; i.e., one that labels unseen examples as "benign" or "malicious" with high precision and recall. However, no prior published work considers what comes next: what does the service actually do after it detects abuse? In this paper, we argue that detection as described in previous work is not the goal of those who are fighting OSN abuse. Rather, we believe the goal to be selecting actions (e.g., ban the user, block the request, show a CAPTCHA, or "collect more evidence") that optimize a tradeoff between harm caused by abuse and impact on benign users. With this framing, we see that enlarging the set of possible actions allows us to move the Pareto frontier in a way that is unattainable by simply tuning the threshold of a binary classifier. To demonstrate the potential of our approach, we present Predictive Response Optimization (PRO), a system based on reinforcement learning that utilizes available contextual information to predict future abuse and user-experience metrics conditioned on each possible action, and select actions that optimize a multi-dimensional tradeoff between abuse/harm and impact on user experience. We deployed versions of PRO targeted at stopping automated activity on Instagram and Facebook. In both cases our experiments showed that PRO outperforms a baseline classification system, reducing abuse volume by 59% and 4.5% (respectively) with no negative impact to users. We also present several case studies that demonstrate how PRO can quickly and automatically adapt to changes in business constraints, system behavior, and/or adversarial tactics.
... Note that for (a)-(c), the model order n is identical to the dimension of the regressor space, while in (d), the regressor space is 2n-dimensional. From linear system identification, it is well known that ARX models need less high orders to well approximate a dynamic process compared to FIR models, [17]. Thus, the regressor space can be chosen as lower dimensional. ...
System identification can be used to determine data-driven mathematical models of dynamic processes. For nonlinear processes, model architectures that are as flexible as possible are required. One possibility is to utilize Gaussian processes (GPs) as a universal approxi-mator with an external dynamics realization, leading to highly flexible models. Novel Laguerre and Kautz filter-based dynamics realizations in GP models are proposed. The Laguerre/Kautz pole(s) are treated as hyperparame-ters with the GPs' standard hyperparameter for the SE-ARD kernel. The two novel dynamics realizations in GP models are compared to different state-of-the-art dynamics realizations such as finite impulse response (FIR) or autoregressive with exogenous input (ARX). The big data case is handled via support points. Using Laguerre and Kautz regressor spaces allows both the dimensionality of the regressor space to be kept small and achieve superior performance. This is demonstrated through numerical examples and measured benchmark data of a Wiener-Hammerstein process.
Accurately forecasting CO2 emissions in the transportation sector is essential for developing effective mitigation strategies. This study uses an annually spanning dataset from 1993 to 2022 to evaluate the predictive performance of three methods: NAR, NARX, and GA-T2FIS. Among these, NARX-VK, which incorporates vehicle kilometers (VK) and economic variables, demonstrated the highest predictive accuracy, achieving a MAPE of 2.2%, MAE of 1621.449 × 103 tons, and RMSE of 1853.799 × 103 tons. This performance surpasses that of NARX-RG, which relies on registered vehicle data and achieved a MAPE of 3.7%. While GA-T2FIS exhibited slightly lower accuracy than NARX-VK, it demonstrated robust performance in handling uncertainties and nonlinear relationships, achieving a MAPE of 2.6%. Sensitivity analysis indicated that changes in VK significantly influence CO2 emissions. The Green Transition Scenario, assuming a 10% reduction in VK, led to a 4.4% decrease in peak CO2 emissions and a 4.1% reduction in total emissions. Conversely, the High Growth Scenario, modeling a 10% increase in VK, resulted in a 7.2% rise in peak emissions and a 4.1% increase in total emissions.
This study introduces an innovative approach to urban waste management in Kerman, Iran, by developing a small-scale digital twin and a fuzzy logic decision support system. The primary goal is to bridge the gap between formal and informal recycling sectors and integrate informal collectors into the formal system. The model, created in MATLAB, integrates both sectors, providing a comprehensive view of stakeholder interactions. By combining "modeling" and "system identification" techniques, the model accurately represents the system's dynamics, validated with historical data. A fuzzy logic controller was designed to optimize competitive pricing strategies for recyclable materials and licensing fees, enabling the municipality to compete effectively with informal recycling networks. The digital twin framework provides a comprehensive model for simulating various scenarios. This approach allows policymakers to assess the impact of different pricing and licensing strategies on municipal revenue, participation rates, and overall system efficiency. The results demonstrate that financial licensing and performance-based discounts can significantly enhance the municipality's market share and revenue. This research demonstrates the potential of digital twin technology in creating resilient and responsive waste management systems, offering valuable insights for similar urban challenges.
Many natural motor skills, such as speaking or locomotion, are acquired through a process of trial-and-error learning over the course of development. It has long been hypothesized, motivated by observations in artificial learning experiments, that dopamine has a crucial role in this process. Dopamine in the basal ganglia is thought to guide reward-based trial-and-error learning by encoding reward prediction errors¹, decreasing after worse-than-predicted reward outcomes and increasing after better-than-predicted ones. Our previous work in adult zebra finches—in which we changed the perceived song quality with distorted auditory feedback—showed that dopamine in Area X, the singing-related basal ganglia, encodes performance prediction error: dopamine is suppressed after worse-than-predicted (distorted syllables) and activated after better-than-predicted (undistorted syllables) performance². However, it remains unknown whether the learning of natural behaviours, such as developmental vocal learning, occurs through dopamine-based reinforcement. Here we tracked song learning trajectories in juvenile zebra finches and used fibre photometry³ to monitor concurrent dopamine activity in Area X. We found that dopamine was activated after syllable renditions that were closer to the eventual adult version of the song, compared with recent renditions, and suppressed after renditions that were further away. Furthermore, the relationship between dopamine and song fluctuations revealed that dopamine predicted the future evolution of song, suggesting that dopamine drives behaviour. Finally, dopamine activity was explained by the contrast between the quality of the current rendition and the recent history of renditions—consistent with dopamine’s hypothesized role in encoding prediction errors in an actor–critic reinforcement-learning model4,5. Reinforcement-learning algorithms⁶ have emerged as a powerful class of model to explain learning in reward-based laboratory tasks, as well as for driving autonomous learning in artificial intelligence⁷. Our results suggest that complex natural behaviours in biological systems can also be acquired through dopamine-mediated reinforcement learning.
The robustness analysis for neural networks aims at evaluating the influence on accuracy induced by perturbations affecting the computational flow; as such it allows the designer for estimating the resilience of the neural model w.r.t perturbations. In the literature, the robustness analysis of neural networks generally focuses on the effects of perturbations affecting biases and weights. The study of the network’s parameters is relevant both from the theoretical and the application point of view, since free parameters characterize the “knowledge space” of the neural model and, hence, its intrinsic functionality. A robustness analysis must also be taken into account when implementing a neural network (or the intelligent computational system into which a neural network is inserted) in a physical device or in intelligent wireless sensor networks. In these contexts, perturbations affecting the weights of a neural network abstract uncertainties such as finite precision representations, fluctuations of the parameters representing the weights in analog solutions (e.g., associated with the production process of a physical component), ageing effects or more complex, and subtle uncertainties in mixed implementations.
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
This article deals with the design of a generalized methodology for the identification of nonlinear dynamic systems using a combination of analytical and experimental identification methods. The proposed methodology is divided into modules that can be repeated if necessary. Verification of the methodology is presented on the modified helicopter educational model CE 150 from Humusoft company. A mathematical model of the system with both rotors approximated by a black-box nonlinear model is presented. The proposed experiments for obtaining experimental data are described in detail and are used in the estimation of model parameters. The gray-box model is validated in a control structure with a control law optimized for microcontrollers. Indirect validation of the gray-box model with the real helicopter educational model points to good approximation properties of the gray-box model and its potential for further use in the identification and control using artificial intelligence methods.
A state-of-the-art review is presented of mathematical manoeuvring models for surface ships and parameter estimation methods that have been used to build mathematical manoeuvring models for surface ships. In the first part, the classical manoeuvring models, such as the Abkowitz model, MMG, Nomoto and their revised versions, are revisited and the model structure with the hydrodynamic coefficients is also presented. Then, manoeuvring tests, including both the scaled model tests and sea trials, are introduced with the fact that the test data is critically important to obtain reliable results using parameter estimation methods. In the last part, selected papers published in journals and international conferences are reviewed and the statistical analysis of the manoeuvring models, test data, system identification methods and environmental disturbances used in the paper is presented.
This paper discusses model validation of hydraulic joints in the presence of unknown external forces using ready-made components. First, we review hydraulic cylinder dynamics with nonlinear pressure dynamics. Second, we design a controller that applies existing methods so that hydraulic joints have high backdrivability against unknown external forces. Third, we discuss the experimental and numerical responses of hydraulic joints in the presence of unknown external forces as the model validation. No custom-made cylinders, valves, or pipes are used in our experiments. Remarkably, the experimental and numerical responses were close to each other in the presence of unknown external forces.
A wavelet network concept, which is based on wavelet transform theory, is proposed as an alternative to feedforward neural networks for approximating arbitrary nonlinear functions. The basic idea is to replace the neurons by ;wavelons', i.e., computing units obtained by cascading an affine transform and a multidimensional wavelet. Then these affine transforms and the synaptic weights must be identified from possibly noise corrupted input/output data. An algorithm of backpropagation type is proposed for wavelet network training, and experimental results are reported.
The problem of the approximation of nonlinear mapping, (especially
continuous mappings) is considered. Regularization theory and a
theoretical framework for approximation (based on regularization
techniques) that leads to a class of three-layer networks called
regularization networks are discussed. Regularization networks are
mathematically related to the radial basis functions, mainly used for
strict interpolation tasks. Learning as approximation and learning as
hypersurface reconstruction are discussed. Two extensions of the
regularization approach are presented, along with the approach's
corrections to splines, regularization, Bayes formulation, and
clustering. The theory of regularization networks is generalized to a
formulation that includes task-dependent clustering and dimensionality
reduction. Applications of regularization networks are discussed
A technique for numerical identification of a discrete time system from input/output samples is described. The purpose of the identification is to design strategies for control of the system. The strategies are obtained using linear stochastic control theory.
A technique for numerical identification of a discrete time system from input/output samples is described. The purpose of the identification is to design strategies for control of the system. The strategies are obtained using linear stochastic control theory.
The parameters of the system are estimated by Maximum Likelihood. An algorithm for solving the M. L. equations is given. The estimates are in general consistent, asymptotically normal and efficient for increasing sample lengths. These properties and also the parameter accuracy are determined by the information matrix. An estimate of this matrix is given.
The technique has been applied to simulated data and to plant data.
Very general reduced order filtering and modeling problems are phased in terms of choosing a state based upon past information to optimally predict the future as measured by a quadratic prediction error criterion. The canonical variate method is extended to approximately solve this problem and give a near optimal reduced-order state space model. The approach is related to the Hankel norm approximation method. The central step in the computation involves a singular value decomposition which is numerically very accurate and stable. An application to reduced-order modeling of transfer functions for stream flow dynamics is given.
This paper is a study of the problem of how to parametrize the set of all finite order constant linear systems. The parameters are interpreted as independent invariants for the equivalence relation which defines two systems to be equivalent when they have the same impulse response. Two kinds of canonical representations of the systems are constructed from the invariants, one of the state-space equations type and the other of the transfer function type.RésuméCe texte fournit une étude du problème de paramétrisation de l'ensemble de tous les systemès linéaires constants d'ordre défini. Les paramètres sont interprêtés comme invariantes indépendantes pour la relation d'équivalence qui définit que deux systèmes sont équivalents lorsqu'ils ont la même réponse d'impulsion. Deux sortes de représentations canoniques des systèmes sont construits à partir des invariantes, l'une du type équation état-espace et l'autre du type fonction de transfert.ZusammenfassungBetrachet wird das Problem, wie die Menge aller konstanten linearen Systeme endlicher Ordnung zu parametrisieren sind. Die Parameter werden als unabhängige Invariante für die Aquivalenzrelation interpretiert, die definiert, daß zwei Systeme äquivalent sind, wenn sie die gleiche Impulsantwort besitzen. Aus den Invarianten wurden zwei Arten kanonischer Darstellung der Systeme konstruiert, eine vom Typ der Gleichungen für den Zustandsraum und die andere vom Typ der Übertragungsfunktion.РефератИзyчaeтcя пpoбнeмa пapaмeтpизaции нaбopa вceч кoнeчнopaзмepныч линeйныч cиcтeм. Пapaмeтpы paccмaтpивaютcя кaк нeзaвиcимo инвapиaнтныe для эквивaлeнтнoгo oтнoшeния, кoтopoe oпpeдeляeт двe cиcтeмы кaк oдинaкoвыe, ecли oни имeют oдинaкoвыe чacтoтныe peaкции. Кoнcтpyктиpyютcя двa видa кaнoничecкич пpeдcтaвлeний cиcтeм из инвapиaнтoв—oднo типa ypaвнeний фaзoвoгo пpocтpaнcтвa и дpyгoe—типa пepeдaтoчнoй фyнкции.
The number of digits it takes to write down an observed sequence x1, …, xN of a time series depends on the model with its parameters that one assumes to have generated the observed data. Accordingly, by finding the model which minimizes the description length one obtains estimates of both the integer-valued structure parameters and the real-valued system parameters.
An outstanding introduction to the fundamentals of regression analysis-updated and expanded The methods of regression analysis are the most widely used statistical tools for discovering the relationships among variables. This classic text, with its emphasis on clear, thorough presentation of concepts and applications, offers a complete, easily accessible introduction to the fundamentals of regression analysis. Assuming only a basic knowledge of elementary statistics, Applied Regression Analysis, Third Edition focuses on the fitting and checking of both linear and nonlinear regression models, using small and large data sets, with pocket calculators or computers. This Third Edition features separate chapters on multicollinearity, generalized linear models, mixture ingredients, geometry of regression, robust regression, and resampling procedures. Extensive support materials include sets of carefully designed exercises with full or partial solutions and a series of true/false questions with answers. All data sets used in both the text and the exercises can be found on the companion disk at the back of the book. For analysts, researchers, and students in university, industrial, and government courses on regression, this text is an excellent introduction to the subject and an efficient means of learning how to use a valuable analytical tool. It will also prove an invaluable reference resource for applied scientists and statisticians.
A hinge function y = h ( x ) consists of two
hyperplanes continuously joined together at a hinge. In regression
(prediction), classification (pattern recognition), and noiseless
function approximation, use of sums of hinge functions gives a powerful
and efficient alternative to neural networks with computation times
several orders of magnitude less than is obtained by fitting neural
networks with a comparable number of parameters. A simple and effective
method for finding good hinges is presented
The history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (-2)log-(maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.
In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.
In this paper we discuss several aspects of the mathematical foundations of non-linear black-box identification problem. As we shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger is the number of parameters used to describe the model, more flexible would be the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is a simple fact that good approximation technique can be a basis of good identification algorithm. From this point of view we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and "neuron" approximations and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretic developments for the ...
A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes, as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a user's perspective. It focuses on what are the common features in the different approaches, the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques. It is pointed out that the nonlinear structures can be seen as a concatenation of a mapping from observed data to a regression vector and a nonlinear mapping from the regressor space to the output space. These mappings are discussed separately. The latter mapping is usually formed as a basis...