Nonlinear Output Error Research Articles

Actuator FDI Scheme for a Wind Turbine Benchmark Using Sliding Mode Observers

This paper proposes a fault detection and isolation (FDI) scheme for a wind turbines subject to actuator faults in both the pitch system and the drive train system. The proposed scheme addresses fault detection and isolation problems using a fault estimation approach. The proposed approach considers the use of a particular class of sliding mode observers (SMOs) designed to maintain the sliding motion even in the presence of actuator faults. The fault detection problem is solved by reconstructing the actuator faults through an appropriate analysis of the nonlinear output error injection signal, which is required to keep the SMO in a sliding motion. To ensure accurate fault reconstruction, only two conditions are required, namely that the faults are bounded and they meet the matching condition. A scheme based on a bank of SMOs is proposed to solve the fault detection and isolation problem in the pitch system. For the drive train system, a scheme using only one SMO is proposed. The performance of the proposed FDI scheme is validated by using a wind turbine benchmark model subjected to several actuator faults. Normalized root mean square error (NRMSE) analysis is performed to evaluate the accuracy of the actuator fault estimations.

Joint two‐stage multi‐innovation recursive least squares parameter and fractional‐order estimation algorithm for the fractional‐order input nonlinear output‐error autoregressive model

SummaryThis paper mainly investigates the issue of parameter identification for the fractional‐order input nonlinear output error autoregressive (IN‐OEAR) model. In order to avoid the problem of large computation of redundant parameter estimation, the output form of the system can be expressed by a linear combination of unknown parameters through the key term separation. Through employing the hierarchial identification principle, the fractional‐order IN‐OEAR model is decomposed into two sub‐models with a smaller number of parameters. On the basis of the recursive identification methods, a recursive least squares sub‐algorithm and a gradient stochastic sub‐algorithm are proposed to estimate the parameters and the fractional‐order, respectively. With the aim of achieving more accurate parameter estimates, a two‐stage multi‐innovation least recursive algorithm is proposed by means of the multi‐innovation identification theory. The numerical simulation results test the effectiveness of the proposed methods.

Thermodynamic based robust regulation of a heat exchanger

In this work, a robust regulation problem based on thermodynamic principles is developed to control a heat exchanger. The system is regulated through a nonlinear output error dependent on the total entropy production within the heat exchanger; this error is a function of the output variable and its reference, but it can also be affected by parametric uncertainties and disturbances, so a geometric controller with a high-gain observer and an anti-windup scheme is proposed. Finally, the numerical simulation of the controller shows compliance with stability and robustness criteria.

Design of auxiliary model based normalized fractional gradient algorithm for nonlinear output-error systems

A new avenue of fractional calculus applications has emerged that investigates the design of fractional gradient based novel iterative methods for analyzing fractals and nonlinear dynamics in solving engineering and applied sciences problems. The most discussed algorithm in this regard is fractional least mean square (FLMS) algorithm. This study presents an auxiliary model based normalized variable initial value FLMS (AM-NVIV-FLMS) algorithm for input nonlinear output error (INOE) system identification. First, NVIV-FLMS is presented to automatically tune the learning rate parameter of VIV-FLMS and then the AM-NVIV-FLMS is introduced by incorporating the auxiliary model idea that replaces the unknown values of the information vector with the output of auxiliary model. The proposed AM-NVIV-FLMS scheme is accurate, convergent, robust and reliable for INOE system identification. Simulation results validate the significance and efficacy of the proposed scheme.

PARAMETER ESTIMATION OF NONLINEAR OUTPUT ERROR SYSTEM UNDER VARIATIONAL BAYESIAN METHOD BASED ON PROBABILISTIC GRAPHICAL MODEL

To estimate the parameters of the nonlinear output error system (nonlinear system), a variational Bayesian estimation method (VB method) is proposed based on the probabilistic graphical model (PGM). First, the related theories are introduced in this study such as the PGM and nonlinear systems. Then, the parameter estimation model of the nonlinear system is established. Finally, a VB method is proposed based on the PGM to estimate the parameters of the nonlinear system, which is tested and verified by numerical simulation experiments. It is found that the parameter estimation model of nonlinear system and the proposed method can estimate the parameters of relevant nonlinear system better, and the minimum error between the parameter estimated value and the actual value is only about 0.0001. It proves the feasibility of the VB method based on PGM in the identification of nonlinear systems. The results of this study provide an important reference for the control and identification of nonlinear systems.

A long short-term memory based Quasi-Virtual Analyzer for dynamic real-time soft sensing of a Simulated Moving Bed unit

The evaluation of unmeasurable quantities is an issue faced in several fields. One example is in the production of pharmaceuticals, in which, typically, isomer properties related to different effects are found: one enantiomer may kill while the other may cure. Thus, strict quality control is necessary in this field, which requires accurate, reliable and frequent measurements of the system. However, the measurement of the main quality properties associated with the production of some pharmaceuticals has a low frequency. Consequently, in addition to safety-related issues, financial losses are also related to these low frequency of measurements, as the product can easily run towards an out of spec production. In this scenario, the present work proposes a novel Deep Artificial Intelligence structure, which has an intrinsic (Nonlinear Output Error) NOE structure, associated with a (Nonlinear AutoRegressive with Exogenous input) NARX predictor, to be used as an online soft sensor in order to provide information about the main properties of a Simulated Moving Bed chromatographic unit, commonly used in the production of pharmaceuticals, in order to mitigate the low frequency of measurement associated with this unit. The proposed structure is here called Improved Quasi-Virtual Analyzer. The model can adapt itself as the process evolves, having the possibility of online learning through measurements obtained in the laboratory periodically. The proposed structure was tested in a software-in-the-loop scenario and compared with a more traditional alternative. These tests showed a robust capacity of the Improved Quasi-Virtual Analyzer to provide reliable predictions in real-time, as well as to outperform the traditional artificial network structure.

Maximum-Likelihood-Based Adaptive and Intelligent Computing for Nonlinear System Identification

Most real-time systems are nonlinear in nature, and their optimization is very difficult due to inherit stiffness and complex system representation. The computational intelligent algorithms of evolutionary computing paradigm (ECP) effectively solve various complex, nonlinear optimization problems. The differential evolution algorithm (DEA) is one of the most important approaches in ECP, which outperforms other standard approaches in terms of accuracy and convergence performance. In this study, a novel application of a recently proposed variant of DEA, the so-called, maximum-likelihood-based, adaptive, differential evolution algorithm (ADEA), is investigated for the identification of nonlinear Hammerstein output error (HOE) systems that are widely used to model different nonlinear processes of engineering and applied sciences. The performance of the ADEA is evaluated by taking polynomial- and sigmoidal-type nonlinearities in two case studies of HOE systems. Moreover, the robustness of the proposed scheme is examined for different noise levels. Reliability and consistent accuracy are assessed through multiple independent trials of the scheme. The convergence, accuracy, robustness and reliability of the ADEA are carefully examined for HOE identification in comparison with the standard counterpart of the DEA. The ADEA achieves the fitness values of 1.43 × 10−8 and 3.46 × 10−9 for a population size of 80 and 100, respectively, in the HOE system identification problem of case study 1 for a 0.01 nose level, while the respective fitness values in the case of DEA are 1.43 × 10−6 and 3.46 × 10−7. The ADEA is more statistically consistent but less complex when compared to the DEA due to the extra operations involved in introducing the adaptiveness during the mutation and crossover. The current study may consider the approach of effective nonlinear system identification as a step further in developing ECP-based computational intelligence.

Hierarchical gradient- and least squares-based iterative algorithms for input nonlinear output-error systems using the key term separation

This paper considers the parameter identification problems of the input nonlinear output-error (IN-OE) systems, that is the Hammerstein output-error systems. In order to overcome the excessive calculation amount of the over-parameterization method of the IN-OE systems. Through applying the hierarchial identification principle and decomposing the IN-OE system into three subsystems with a smaller number of parameters, we present the key term separation auxiliary model hierarchical gradient-based iterative algorithm and the key term separation auxiliary model hierarchical least squares-based iterative algorithm, which are called the key term separation auxiliary model three-stage gradient-based iterative algorithm and the key term separation auxiliary model three-stage least squares-based iterative algorithm. The comparison of the calculation amount and the simulation analysis indicate that the proposed algorithms are effective.

Intake O <sub>2</sub> Concentration Estimation in a Turbocharged Diesel Engine through NOE

<div class="section abstract"><div class="htmlview paragraph">Diesel engines with their embedded control systems are becoming increasingly complex as the emission regulations tighten, especially concerning NO<sub>x</sub> pollutants. The combustion and emission formation processes are closely correlated to the intake manifold O<sub>2</sub> concentration. Consequently, the performance of the engine controllers can be improved if a model-based or sensor-based estimation of the O<sub>2</sub> concentration is available. The paper addresses the modeling of the O<sub>2</sub> concentration in a turbocharged diesel engine. Dynamic models, compared to generally employed steady state maps, capture the dynamic effects occurring over transients, when the major deviations from the stationary maps are found. Dynamic models positively affect the control system making it more effective and, exploiting information coming from sensors, they provide a more robust prediction performance. Firstly, a Nonlinear Output Error model (NOE), with simulation focus, fed with four inputs is presented. The considered nonlinear function set is the one of neural networks. The inputs are engine BMEP, engine RPM and EGR and VGT valves position. Two distinct datasets are used for training and validation of the NOE model. These sets are generated using GT-Power simulation software implementing a fine model of the engine, previously validated on experimental measurements taken on the real engine. Besides the transient validation, the NOE model was tested against GT-Power outputs on step tests involving the EGR and VGT actuators. At last the network output is compared with an O<sub>2</sub> steady state map over a transient in normal and faulty conditions. The performance of the model is satisfactory in both conditions. Secondly, the potential benefits of installing an O<sub>2</sub> sensor in the intake manifold is presented: a Nonlinear Auto-Regressive with eXogenous input (NARX) model is considered and compared to the previously investigated NOE. The results prove that, exploiting the output coming from the O<sub>2</sub> sensor, the model prediction capability significantly improves.</div></div>

Optimizing long short-term memory recurrent neural networks using ant colony optimization to predict turbine engine vibration

This article expands on research that has been done to develop a recurrent neural network (RNN) capable of predicting aircraft engine vibrations using long short-term memory (LSTM) neurons. LSTM RNNs can provide a more generalizable and robust method for prediction over analytical calculations of engine vibration, as analytical calculations must be solved iteratively based on specific empirical engine parameters, making this approach ungeneralizable across multiple engines. In initial work, multiple LSTM RNN architectures were proposed, evaluated and compared. This research improves the performance of the most effective LSTM network design proposed in the previous work by using a promising neuroevolution method based on ant colony optimization (ACO) to develop and enhance the LSTM cell structure of the network. A parallelized version of the ACO neuroevolution algorithm has been developed and the evolved LSTM RNNs were compared to the previously used fixed topology. The evolved networks were trained on a large database of flight data records obtained from an airline containing flights that suffered from excessive vibration. Results were obtained using MPI (Message Passing Interface) on a high performance computing (HPC) cluster, evolving 1000 different LSTM cell structures using 208 cores over 21 days. The new evolved LSTM cells showed an improvement of 1.34%, reducing the mean prediction error from 5.61% to 4.27% when predicting excessive engine vibrations 10 s in the future, while at the same time dramatically reducing the number of weights from 21,170 to 13,150. The optimized LSTM also performed significantly better than traditional Nonlinear Output Error (NOE), Nonlinear AutoRegression with eXogenous (NARX) inputs, and Nonlinear Box–Jenkins (NBJ) models, which only reached error rates of 15.73%, 12.06% and 15.05%, respectively. Furthermore, the LSTM regularization method was used to validate the ACO. The ACO LSTM out performed the regularized LSTM by 3.35%. The NOE, NARX, and NBJ models were also regularized for cross validation, and the mean prediction errors were 8.70%, 9.40%, and 9.43% respectively, which gives credit for the ant colony optimized LSTM RNN.

Stabilization of Switched Nonlinear Systems by Adaptive Observer-Based Dynamic Surface Control with Nonlinear Virtual and Output Feedback

This paper presents a new nonlinear output and virtual error feedback control technique for stabilization of switched uncertain nonlinear systems using a continuous differentiable nonlinear feedback function (NFF). The merits of the proposed method are threefold: (i) there is no assumption that the switching signals must satisfy the (average) dwell time, (ii) the feedback amplitude self-adjusts under different state levels to guarantee better dynamic performance using the NFF, and (iii) the number of parameters that needs online tuning is 1, i.e., the proposed control method is computationally inexpensive. Moreover, the closed-loop signals are kept bounded using the rigorously proved Lyapunov and Invariant-set theorems, and the output signal converges to a sufficiently small region around zero by choosing proper design parameters. Finally, simulation and comparative results are given to demonstrate the effectiveness of the proposed method.

Exponential State Observers for Nonlinear Systems with Incremental Quadratic Constraints and Output Nonlinearities

This paper considers the exponential observer design for a class of nonlinear systems with output nonlinearities. The nonlinear terms in the systems are assumed to satisfy incremental quadratic constraints which include many commonly encountered nonlinearities in existing literature as some special cases. We construct a circle-criterion-based observer by injecting both the linear and the nonlinear output error terms into the observer system. Sufficient conditions ensuring the exponential stability of the proposed observer are established and formulated in terms of linear matrix inequalities. Finally, the advantages and effectiveness of the proposed design approach are illustrated through two examples.

Speed-up of Iterative Real-Time Optimization by Estimating the Steady States in the Transient Phase using Nonlinear System Identification

Iterative Real-Time Optimization (RTO) has gained increasing attention in the context of model-based optimization of the operating points of chemical plants in the presence of plant-model mismatch. In all iterative RTO schemes, it is necessary to wait until the plant has reached a steady-state to obtain the required information on plant performance and constraint satisfaction which leads to slow convergence in the case of processes with slow dynamics. It has recently been proposed to use a linear black-box model that is identified online to predict the steady-state values of the plant during the transient between different stationary operating points; these values are then employed in the modifier adaptation with quadratic approximation to drive the process to its optimum (Gao et al., 2016). In this contribution, this idea is extended by integrating nonlinear system identification into iterative RTO. Specifically, a Nonlinear Output Error (NOE) model is proposed to describe the dynamics of the process, thus providing a faster prediction of the steady-state of the plant. A robust scheme for the estimation of the model parameters is proposed. The performance of the strategy is illustrated by simulation studies of a continuous stirred-tank reactor. By means of the proposed methodology a fast convergence to the plant optimum can be achieved despite plant-model mismatches.

Partially coupled extended stochastic gradient algorithm for nonlinear multivariable output error moving average systems

PurposeThe purpose of this paper is to study the parameter estimation problem of nonlinear multivariable output error moving average systems.Design/methodology/approachA partially coupled extended stochastic gradient algorithm is presented for nonlinear multivariable systems by using the decomposition technique.FindingsThe proposed algorithm can realize the coupled computation of the parameter estimates between subsystems.Originality/valueThis paper develops a coupled parameter estimation algorithm for nonlinear multivariable systems and directly estimates the system parameters without over-parameterization.

NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data

An approach to Nonlinear Output Error (NOE) modelling using Takagi–Sugeno (TS) fuzzy model for a class of nonlinear dynamic systems having variability in their outputs is presented. Furthermore, the approach is compared and graphically illustrated with other alternate approaches on the basis of interval data and interval membership functions. Assuming the identification method can be repeated offline a number of times under similar conditions, multiple input–output time series can be obtained from the underlying system. These time series are pre-processed using the techniques of statistics and probability theory to generate the envelopes of response (curves outlining the upper and lower extremes of response) at each time instant. Two types of envelopes are described in this research: the max–min envelopes and the envelopes based on the confidence intervals provided by extended Chebyshev's inequality. By incorporating interval data in fuzzy modelling and using the theory of symbolic interval-valued data, a TS fuzzy model with interval antecedent and consequent parameters is obtained. This algorithm provides a model for predicting the expected response as well as envelopes. In order to validate the presented model, a simulation case study is devised in this paper. Moreover, it is demonstrated on the real data obtained from an electro-mechanical throttle valve.

Zero dynamics and funnel control of general linear differential-algebraic systems

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the zero dynamics and tracking control. We introduce and characterize the concept of autonomous zero dynamics as an important system theoretic tool for the analysis of differential-algebraic systems. We use the autonomous zero dynamics and ( E, A, B )-invariant subspaces to derive the so called zero dynamics form – which decouples the zero dynamics of the system – and exploit it for the characterization of system invertibility and asymptotic stability of the zero dynamics. A refinement of the zero dynamics form is then used to show that the funnel controller (that is a static nonlinear output error feedback) achieves – for a special class of right-invertible systems with asymptotically stable zero dynamics – tracking of a reference signal by the output signal within a pre-specified performance funnel. It is shown that the results can be applied to a class of passive electrical networks.

Iterative identification algorithms for input nonlinear output error autoregressive systems

This paper focuses on the parameter estimation problems of input nonlinear output error autoregressive systems. Based on the key variables separation technique and the auxiliary model identification idea, the output of the system is expressed as a linear combination of all the system parameters, the unknown inner variables in the information vector are replaced with the outputs of the auxiliary model and a gradient based and a least squares based iterative identification algorithms are derived. Simulation example is provided to illustrate the effectiveness of the proposed algorithms.

A Control-Oriented Model of a PEM Fuel Cell Stack Based on NARX and NOE Neural Networks

Hydrogen-related technologies have been proposed as an alternative to store the energy surplus from renewable sources. Among these technologies, the proton exchange membrane fuel cell (PEMFC) and electrolyzer are the preferred choice for practical applications since they have reached a certain level of maturity and are commercially available at present. In order to achieve a cost-effective operation, a PEMFC stack must operate at maximum efficiency most of the time. Since PEMFC stacks present a time-varying behavior, an adaptive model-based controller should be employed to accomplish this goal. A fixed-parameter electrochemical model may not offer a reliable prediction over a midterm time horizon for such a controller. For this reason, system identification techniques appear as more appropriate choices to obtain an effective model for this class of control systems. In this paper, a system identification modeling methodology employing nonlinear autoregressive with exogenous input (NARX) and nonlinear output error (NOE) neural networks is presented to obtain a black-box model of a PEMFC stack oriented for a predictive control system. The experimental data for the model development are obtained with a commercial 3-kW PEMFC stack. The model built according to the proposed methodology provides accurate predictions of the voltage for the whole operating range of the stack for a long time and, hence, the ability to represent the time-varying behavior of a PEMFC stack for a predictive control application.

Data Filtering-Based Multi-innovation Stochastic Gradient Algorithm for Nonlinear Output Error Autoregressive Systems

This paper discusses the parameter estimation problems of nonlinear output error autoregressive systems and presents a data filtering-based multi-innovation stochastic gradient algorithm for improving the parameter estimation accuracy of the stochastic gradient algorithm by combining the multi-innovation identification theory and the data filtering technique. The proposed algorithm is effective and can generate highly accurate parameter estimates compared with the multi-innovation stochastic gradient algorithm. The simulation results confirm this conclusion.

A Novel Approach to T-S Fuzzy Modeling of Nonlinear Dynamic Systems with Uncertainties using Symbolic Interval-Valued Outputs

A novel approach to Takagi-Sugeno (T-S) fuzzy modeling of a class of nonlineardynamic systems having variability in their outputs for the Nonlinear Output Error (NOE)case is addressed in this article. Multiple input-output datasets were obtained by repeating the identification experiment. The variability in the output time series is captured by defining the envelops of response at each time instant. These envelops actually provide the confidence interval based upper and lower bounds of the output time series using the extended Chebyshev’s Inequality. Different from the previous approach, in which two independent T-S fuzzy models were used for identifying each bound, a single T-S fuzzy model is identified in this work, which resulted in interval parameters for the antecedent and consequent variables. This is accomplished by first transforming the bounds into the symbolic interval-valued data and then using this data for identification. In order to get the expected value of the response, the estimated lower and upper bound time series of the identified T-S fuzzy model were averaged out at each time instant, as permitted by the extended Chebyshev’s Inequality. The proposed approach is demonstrated on an industrial diesel-engine electro-mechanical throttle valve.