Output Error Model Research Articles

Improvement Instrumental Variables Method for Output Error Model

The identification of parameters system can be done in multiple ways by alternating methods (Least Squares, Instrumental Variables) and models (ARX, OE). The Ordinary Least Squares method which was presented in several research papers gives biased estimates for Output Error model. For example, the interpretation results in the first order OE model, the autocorrelation function shows that the residuals are not white noise, and the LS estimator is well biased, and it is not a suitable estimator for identification of output error model. In this paper, we will present a detailed study of Recursive Instrumental Variables method and its instruments choice to identify the parameters of Output Error model with unbiased estimates. The idea is simultaneously to identify the parameters system similar to an ARX model with Least Squares method. It is a successfully applied to identify with unbiased estimates parameters of a second order system based on whitening error prediction, and we showed that, if the instruments are weak, then the estimator may be biased and confidence intervals and hypothesis tests unreliable. Also, for every data observation set, the estimates parameters can be compared with those originally generated by MATLAB functions. Finally, a numerical example illustrates the efficiency and performances of the proposed method.

Stable Weighted Multiple Model Adaptive Control of Continuous-Time Plant With Large Parameter Uncertainties

This paper presents a weighted multiple model adaptive control (WMMAC) scheme to deal with large parametric uncertainty of continuous-time plant. In this proposed scheme, each `local' controller is designed according to the mixed-μ-synthesis method to consider small uncertainty of plant parameters and disturbance; the weighting algorithm is directly based on model output errors rather than the residuals generated by multiple Kalman filters as in classical multiple model adaptive control (CMMAC). The closed-loop stability (signal boundedness) and tracking performance of the proposed WMMAC system are proved with the help of virtual equivalent system (VES) concept and methodology.

Frequency domain identification of continuous-time output-error models with time-delay from relay feedback tests

This paper is concerned with identification of continuous-time output-error models with time-delay from relay feedback tests. Conventional methods for solving this problem consist in deriving analytical limit cycle expressions and fitting them to measured shape factors. However, they may fail to handle different limit cycles uniformly, due to the structural differences in the analytical expressions. To overcome this problem, we consider a more general, data-based, parametric identification framework using sampled limit cycle data. A frequency domain method that minimizes the sum of squared output-errors is developed. The proposed method can be of high accuracy, thanks to the periodic input–output signals provided by sustained relay feedback oscillations, which can help to reduce leakage and aliasing errors. Besides, a distinctive merit of the proposed method is that identification of stable and unstable plants can be equally simple. The effectiveness and superiority of the proposed method are demonstrated by means of both theoretical analyses and simulation examples.

Parametric system identification and robust controller design for liquid–liquid heat exchanger system

This study provides a step-by-step analysis of a parametric system identification procedure which provides a singleinput single-output, linear time invariant time-delay model of a liquid-liquid heat exchanger system. Different linear and blockoriented models are used to develop a control relevant identification model. The prediction error method is used to estimate the model parameters. From the simulation results, it can be found that the output error model (linear model) and Hammerstain- Wiener model (block-oriented model) provide the best model validation for a heat exchanger system at 72.23% and 81.5 % goodness-of-FIT, respectively. To achieve the control objectives of the system, the classical proportional-integral-derivative (PID) controller is used. For controller design, the linear model is considered. An industrial process is prone to encounter different uncertainties. Considering the time delay of the estimated linear model as the uncertainty, a ℋ ∞ criterion is used to find out the robust parameters of the PID controller. The developed controller satisfies the sensitivity constraints. Stability boundary locus of (K p ,K i ), (K p ,K d ) and (K d ,K i ) of the PID controller have been plotted to illustrate the robust parameters of the PID controller.

Identification of Hammerstein systems with time delay under load disturbance

To cope with load disturbance often encountered when performing identification tests on non-linear systems with input delay in industrial applications, a bias-eliminated Hammerstein-type output error (OE) model identification method is proposed in this study. By taking into account the load disturbance response as a time-varying parameter for estimation, two recursive least-squares (RLS) identification algorithms is established to estimate the Hammerstein-type model parameters and the time-varying disturbance response. A one-dimensional searching approach is adopted to determine the integer-type delay parameter by minimising the fitting error of output response. Moreover, an auxiliary OE model is constructed to ensure consistent estimation under stochastic noise. In addition, two adaptive forgetting factors are introduced into the proposed RLS algorithms to enhance the estimation convergence on the model parameters and the disturbance response. Asymptotic properties of parameter estimation are analysed along with a proof, in particular for unbiased estimation against a constant disturbance. Two illustrative examples are given to demonstrate the effectiveness of the proposed identification method.

Time-Partitioned Piecewise Affine Output Error Model for Batch Processes

In this paper, a time-partitioned piecewise affine output error (PWA-OE) model for batch processes is proposed, in which time index is used to simplify the partition of regression domain by utilizi...

Correcting the Sea Surface Temperature by Data Assimilation Over the Persian Gulf

In this paper, the impact of sea surface temperature (SST) data assimilation on the results of a finite volume community ocean model has been examined by using the nudging scheme. In this regard, the advanced very high-resolution radiometer satellite SST data were selected as the observational data for assimilation. The numerical modeling was performed over the Persian Gulf from 1998 to 2003 in two different modes: with and without SST data assimilation. The performance of data assimilation with the nudging scheme was evaluated by comparing the simulated SST with the in situ SST measurements and optimum interpolation SST data. Both the spatial and temporal comparisons show the efficiency of assimilation in correcting the model results. The spatial root mean square error in the assimilated run depicts meaningful improvements in the whole of the domain. Also, the temporal comparisons of the results show the capability of assimilation in lowering the model output errors. The simulated SST obtained by applying the data assimilation in the shallow parts of the Persian Gulf matched exactly with the measured ones, especially near the Hormuz Strait. Finally, the results show significant improvements in the SST simulated by using the nudging schemes.

Wavelet based Steglitz Mc-Bride algorithm for Identification of Multiscale Output-Error Models

Identification of output-error models for systems with multiple time scales is known to be a challenging problem due to the spread of dynamics across a wide range of time scales. The wide separation in time constants of the system forces one to deal with unusually fast sampling rates for slow dynamics. In general, output-error model identification of multiscale systems using prediction-error minimization suffers whenever they are initialized with auto-regressive exogenous (ARX) model parameter estimates owing to the sensitivity of ARX models at fast sampling. In the present work, it is observed that the classical Steglitz Mc-Bride (SM) algorithm can serve as an excellent alternative for identification of simple multiscale systems. Despite its advantages, it is seen to yield unstable models at times (related to the sensitivity of ARX estimation) and sometimes converges to a secondary optima. In this work, a multiscale SM algorithm, which respects the multiscale nature of data generating process, is proposed in order to reduce the sensitivity issues arising due to fast sampling. The proposed methodology is observed to yield good results for multiscale systems in terms of obtaining stable models and faster convergence. The performance of the proposed method is demonstrated on three simulation examples and the results are compared with traditional methods.

Multi-innovation based Identification of Output Error Model with Time Delay under Load Disturbance

In this paper, a robust output error model identification method is proposed for industrial processes with time delay subject to load disturbance. By viewing the load disturbance response as a time-variant parameter to be estimated, an extended recursive least-squares (RLS) identification method is established to simultaneously estimate the model parameters and the disturbance response, based on a multi-innovation fitting strategy. The integer type time delay parameter is determined by using a one-dimensional searching approach. An auxiliary model is used to estimate the noise-free output against stochastic noise. Besides, two adaptive forgetting factors are introduced to improve the convergence rates of estimating the time-invariant model parameters and the time-variant disturbance response, respectively. The effectiveness and merit of the proposed method is demonstrated by an illustrative example.

Sparse parameter estimation of LTI models with <i>l<SUP align="right">p</SUP></i> sparsity using genetic algorithm

Sparse optimisation for the identification of parametric linear model structure is equivalent to the estimation of the parameter vector. After relaxing the assumption on the order of the system, sparse optimisation techniques can be utilised to find the optimal model. This paper proposes an optimisation method to find the sparse parameter estimates. For this purpose, lp norm (0 < p < 1) penalty of parameter vector is added to the quadratic loss function which is further minimised using genetic algorithm. For the model structures other than ARX, a simulation model is realised using conditions on the quadratic simulation error. A real coded genetic algorithm is used to minimise the simulation error model. Simulation results are given for the ARX and output error model structures to show the effectiveness of the simulation error model method.

Sparse parameter estimation of LTI models with &lt;i&gt;l&lt;SUP align="right"&gt;p&lt;/SUP&gt;&lt;/i&gt; sparsity using genetic algorithm

Sparse optimisation for the identification of parametric linear model structure is equivalent to the estimation of the parameter vector. After relaxing the assumption on the order of the system, sparse optimisation techniques can be utilised to find the optimal model. This paper proposes an optimisation method to find the sparse parameter estimates. For this purpose, lp norm (0 < p < 1) penalty of parameter vector is added to the quadratic loss function which is further minimised using genetic algorithm. For the model structures other than ARX, a simulation model is realised using conditions on the quadratic simulation error. A real coded genetic algorithm is used to minimise the simulation error model. Simulation results are given for the ARX and output error model structures to show the effectiveness of the simulation error model method.

Enhanced particle swarm optimisation algorithms for multiple-input multiple-output system modelling using convolved Gaussian process models

Convolved Gaussian process (CGP) can capture the input-output correlation, and the correlation of multiple outputs. This is beneficial to the modelling problem of multiple-input multiple-output (MIMO) systems. One key issue of CGP is the learning of hyperparameters from input-output observations. This is typically performed by maximising the log-likelihood (LL) function using gradient based approaches. However, the LL value is not a reliable indicator for judging the quality of intermediate models. We address this issue by minimising the model output error instead. In addition, three enhanced particle swarm optimisation (PSO) algorithms are proposed to solve the optimisation problem because gradient based approaches often get stuck in local optima. The simulation results on numerical linear and nonlinear systems demonstrate the effectiveness of minimising the model output error to learn hyperparameters, and the better performance of using enhanced PSOs compared to gradient based approaches.

Identification of low-order models using Laguerre basis function expansions

In this paper, a method for identification of low-order output-error models expressed in terms of Laguerre basis functions is presented. The identification problem is formulated as a rank-constrained optimization problem in the Laguerre domain, which can be solved efficiently using nuclear norm regularization. Since the identified models are of low order, a minimal state-space realization of the system can be obtained without the use of additional approximative model-order reduction steps.

On Adaptive Boosting for System Identification.

In the field of machine learning, the algorithm Adaptive Boosting has been successfully applied to a wide range of regression and classification problems. However, to the best of the authors' knowledge, the use of this algorithm to estimate dynamical systems has not been exploited. In this brief, we explore the connection between Adaptive Boosting and system identification, and give examples of an identification method that makes use of this connection. We prove that the resulting estimate converges to the true underlying system for an output-error model structure under reasonable assumptions in the large sample limit and derive a bound of the model mismatch for the noise-free case.

Streaming electrification in the swinging plate system

The article presents the results of measurements and analysis of the flow electrification process. The research is presented with a new type of measurement instrument, in which one of the electrodes moves in an oscillating manner. The arrangement comprises two parallel plates and the liquid is relocated in one dimension. One of the electrodes is in the form of a copper wire. As a result, the liquid distribution is assumed as very simple and similar to the classic Stokes boundary layer. The electrification current measurements for different oscillation frequencies were performed in such an arrangement. The basic characteristics of the measured current and the determined process models were tested. The nonlinear approach and the linear approach were used. In the case of the nonlinear mode, the displacement of electrode was employed as an input. Hammerstein's mode with a piecewise linear input nonlinearity estimator was used for identification. In the case of the linear mode, the calculated velocity magnitude was used as an input into the system. The discrete time Output Error model and the Transfer Function model were used as a linear model. In each case, very good compliance was obtained between the simulation responses and the actual current measurements.

Outlier robust stochastic approximation algorithm for identification of MIMO Hammerstein models

This paper considers the robust recursive stochastic gradient algorithm for identification of multivariable Hammerstein model with a static nonlinear block in polynomial form and a linear block described by output-error model. The algorithm is designed for unknown parameters in vector form. It is assumed that there is a priori information about a distribution class to which a real disturbance belongs. Such class of distributions describes the presence of outliers in observations. The main contributions of the paper are: (i) design of robust stochastic approximation algorithm for MIMO Hammerstein models using robust statistics (Huber’s theory); (ii) design of general form of nonlinear block; (iii) a strong consistency of estimated parameter whereby proof is based on martingale theory, generalized strictly positive real condition and persistent excitation condition. The properties of algorithm are illustrated by simulations.

An Adaptive Dual MPC Scheme based on Output Error Models Parameterized using Generalized Orthonormal Basis Filters

A significant fraction of industrial MPC schemes employ linear prediction models. Closed loop performance of a linear model based MPC scheme can deteriorate over a period of time if the prediction model is not updated to account for the changing operating conditions. A possible remedy to this problem is on-line update of the model parameters under the closed loop conditions. An effective way of handling this problem is through dual control, which directs the plant output towards a reference setpoint and simultaneously injects probing signals into the plant to get information-rich data. In this work, an adaptive dual MPC scheme is developed for controlling MIMO systems based on output error models (OE) parameterized using generalized orthogonal basis filters (GOBF). A nominal model is initially developed using o-ine identification exercise. The Fourier coe¢cients of GOBF-OE models are then updated online using recursive least squares algorithm. Similar to Kumar et al. [2015], the MPC formulation is modified to include terms that are sensitive to the parameter covariance and are capable of injecting probing perturbations into the system as and when required. A distinguishing feature of the proposed work is the use of state space realizations of GOBF networks for model development and prediction. Simulation studies using the benchmark quadruple tank system (Johansson [2000]) reveal that the proposed approach provides su¢cient degrees of freedom to excite the plant in closed loop for generating information rich data for model parameter estimation.

Shooting Methods for Parameter Estimation of Output Error Models

This paper studies parameter estimation of output error (OE) models. The commonly used approach of minimizing the free-run simulation error is called single shooting in contrast with the new multiple shooting approach proposed in this paper, for which the free-run simulation error of sub-datasets is minimized subject to equality constraints. The names “single shooting” and “multiple shooting” are used due to the similarities with techniques for estimating ODE (ordinary differential equation) parameters. Examples with nonlinear polynomial models illustrate the advantages of OE models as well as the capability of the multiple shooting approach to avoid undesirable local minima.

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.

Recursive Output Error Identification Algorithm for Switched Linear systems with Bounded Noise

The problem considered in this paper is the identification from input/output data of Switched linear systems with Switched Output Error (SOE) models. The identification is carried out in the case where an upper bound on the noise is assumed to be known. This problem includes the estimation of the discrete state that corresponds to the active sub-model and the estimation of the parameters of all sub-models. The proposed algorithm proceeds in two successive stages. The stability and convergence proprieties are established.