An efficient hierarchical identification method for general dual-rate sampled-data systems

For the lifted input–output representation of general dual-rate sampled-data systems, this paper presents a decomposition based recursive least squares (D-LS) identification algorithm using the hierarchical identification principle. Compared with the recursive least squares (RLS) algorithm, the proposed D-LS algorithm does not require computing the covariance matrices with large sizes and matrix inverses in each recursion step, and thus has a higher computational efficiency than the RLS algorithm. The performance analysis of the D-LS algorithm indicates that the parameter estimates can converge to their true values. A simulation example is given to confirm the convergence results.     

Two recursive least squares parameter estimation algorithms for multirate multiple-input systems by using the auxiliary model

This paper considers identification problems of multirate multiple-input output error systems, derives the input-output representations by using the state space models of the multirate systems, and presents two auxiliary model based recursive least squares algorithms for the corresponding output error models with each subsystem having different or same denominator polynomials. The simulation results show the effectiveness of the proposed algorithms.     

Recursive least squares parameter identification algorithms for systems with colored noise using the filtering technique and the auxilary model

This paper focuses on the parameter estimation problems of output error autoregressive systems and output error autoregressive moving average systems (i.e., the Box–Jenkins systems). Two recursive least squares parameter estimation algorithms are proposed by using the data filtering technique and the auxiliary model identification idea. The key is to use a linear filter to filter the input–output data. The proposed algorithms can identify the parameters of the system models and the noise models interactively and can generate more accurate parameter estimates than the auxiliary model based recursive least squares algorithms. Two examples are given to test the proposed algorithms.     

Auxiliary model-based least-squares identification methods for Hammerstein output-error systems

The difficulty in identification of a Hammerstein (a linear dynamical block following a memoryless nonlinear block) nonlinear output-error model is that the information vector in the identification model contains unknown variables—the noise-free (true) outputs of the system. In this paper, an auxiliary model-based least-squares identification algorithm is developed. The basic idea is to replace the unknown variables by the output of an auxiliary model. Convergence analysis of the algorithm indicates that the parameter estimation error consistently converges to zero under a generalized persistent excitation condition. The simulation results show the effectiveness of the proposed algorithms.     

Outliers, inliers and the generalized least trinuned squares estimator in system identification

The least trimmed squares estimator (LTS) is a well known robust estimator in terms of protecting the estimate from the outliers. Its high computational complexity is however a problem in practice. We show that the LTS estimate can be obtained by a simple algorithm with the complexity O( N In N) for large N, where N is the number of measurements. We also show that though the LTS is robust in terms of the outliers, it is sensitive to the inliers. The concept of the inliers is introduced. Moreover, the Generalized Least Trimmed Squares estimator (GLTS) together with its solution are presented that reduces the effect of both the outliers and the inliers.     

Outliers, inliers and the generalized least trimmed squares estimator in system identification

The least trimmed squares estimator (LTS) is a well known robust estinaator in terms of protecting the estimatefrom the outliers. Its high computational complexity is however a problem in practice. We show that the LTS estimate can be obtained by a simple algorithm with the complexity O( N In N) for large N, where N is the number of measurements. We also showthat though the LTS is robust in terms of the outliers, it is sensitive to the inliers. The concept of the inliers is introduced. Moreover, the Generalized Least Trimmed Squares estimator (GLTS) together with its solution are presented that reduces the effect of both the outliers and the inliers.     

Combined parameter and output estimation of dual-rate systems using an auxiliary model

For a dual-rate sampled-data system, an auxiliary model based identification algorithm for combined parameter and output estimation is proposed. The basic idea is to use an auxiliary model to estimate the unknown noise-free output (true output) of the system, and directly to identify the parameters of the underlying fast single-rate model from the dual-rate input-output data. It is shown that the parameter estimation error consistently converges to zero under generalized or weak persistent excitation conditions and unbounded noise variance, and that the output estimates uniformly converge to the true outputs. An example is included.     

Identification of Hammerstein nonlinear ARMAX systems

Two identification algorithms, an iterative least-squares and a recursive least-squares, are developed for Hammerstein nonlinear systems with memoryless nonlinear blocks and linear dynamical blocks described by ARMAX/CARMA models. The basic idea is to replace unmeasurable noise terms in the information vectors by their estimates, and to compute the noise estimates based on the obtained parameter estimates. Convergence properties of the recursive algorithm in the stochastic framework show that the parameter estimation error consistently converges to zero under the generalized persistent excitation condition. The simulation results validate the algorithms proposed.     

Convergence of normalized iterative identification of Hammerstein systems

An iterative identification algorithm of Hammerstein systems needs a proper initial condition to guarantee its convergence. In this paper, we propose a new algorithm by fixing the norm of the parameter estimates. The normalized algorithm ensures the convergence property under arbitrary nonzero initial conditions. The proofs of the property also give a geometrical explanation on why the normalization guarantees the convergence. An additional contribution is that the static function in the Hammerstein system is extended to square-integrable functions.     

Instability detection of ARMA systems based on AR system identification

The ability of parametric autoregressive (AR) system identification methods to detect the instability of an autoregressive moving average (ARMA) system of an unknown order is investigated. The collection of least squares AR estimators of various orders is shown to have the capacity to detect the instability of the underlying system. Necessary information is not the order of the system but, instead, an upper bound of the number of unstable poles with the maximal magnitude outside the unit circle.     

Performance analysis of multi-innovation gradient type identification methods

It is well-known that the stochastic gradient (SG) identification algorithm has poor convergence rate. In order to improve the convergence rate, we extend the SG algorithm from the viewpoint of innovation modification and present multi-innovation gradient type identification algorithms, including a multi-innovation stochastic gradient (MISG) algorithm and a multi-innovation forgetting gradient (MIFG) algorithm. Because the multi-innovation gradient type algorithms use not only the current data but also the past data at each iteration, parameter estimation accuracy can be improved. Finally, the performance analysis and simulation results show that the proposed MISG and MIFG algorithms have faster convergence rates and better tracking performance than their corresponding SG algorithms.     

Hierarchical gradient-based identification of multivariable discrete-time systems

In this paper, we use a hierarchical identification principle to study identification problems for multivariable discrete-time systems. We propose a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and prove that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation. The proposed algorithms can be applied to identification of systems involving non-stationary signals and have significant computational advantage over existing identification algorithms. Finally, we test the proposed algorithms by simulation and show their effectiveness.     

Data filtering based least squares iterative algorithm for parameter identification of output error autoregressive systems

This letter derives a data filtering based least squares iterative identification algorithm for output error autoregressive systems. The basic idea is to use the data filtering technique to transform the original identification model to an equation error model and to estimate the parameters of this model. The proposed algorithm is more efficient and can produce more accurate parameter estimation than the existing least squares iterative algorithm.     

A random least-trimmed-squares identification algorithm

The Least-trimmed-squares (LTS) estimator is a well known robust estimator in terms of protecting the estimate from the outliers. Its high computational complexity is however a problem in practice. In this paper, we propose a random LTS algorithm which has a low computational complexity that can be calculated a priori as a function of the required error bound and the confidence interval. Moreover, if the number of data points goes to infinite, the algorithm becomes a deterministic one that converges to the true LTS in some probability sense.     

Hierarchical Newton and least squares iterative estimation algorithm for dynamic systems by transfer functions based on the impulse responses

This paper develops a parameter estimation algorithm for linear continuous-time systems based on the hierarchical principle and the parameter decomposition strategy. Although the linear continuous-time system is a linear system, its output response is a highly nonlinear function with respect to the system parameters. In order to propose a direct estimation algorithm, a criterion function is constructed between the response output and the observation output by means of the discrete sampled data. Then a scheme by combining the Newton iteration and the least squares iteration is builded to minimise the criterion function and derive the parameter estimation algorithm. In light of the different features between the system parameters and the output function, two sub-algorithms are derived by using the parameter decomposition. In order to remove the associate terms between the two sub-algorithms, a Newton and least squares iterative algorithm is deduced to identify system parameters. Compared with the Newton iterative estimation algorithm without the parameter decomposition, the complexity of the hierarchical Newton and least squares iterative estimation algorithm is reduced because the dimension of the Hessian matrix is lessened after the parameter decomposition. The experimental results show that the proposed algorithm has good performance.     

Recursive identification of switched ARX systems

We consider the problem of recursively identifying the parameters of a deterministic discrete-time Switched Auto-Regressive eXogenous (SARX) model, under the assumption that the number of models, the model orders and the mode sequence are unknown. The key to our approach is to view the identification of multiple ARX models as the identification of a single, though more complex, lifted dynamical model built by applying a polynomial embedding to the input/output data. We show that the dynamics of this lifted model do not depend on the value of the discrete state or the switching mechanism, and are linear on the so-called hybrid model parameters. Therefore, one can identify the parameters of the lifted model using a standard recursive identifier applied to the embedded input/output data. The estimated hybrid model parameters are then used to build a polynomial whose derivatives at a regressor give an estimate of the parameters of the ARX model generating that regressor. The estimated ARX model parameters are shown to converge exponentially to their true values under a suitable persistence of excitation condition on a projection of the embedded input/output data. Such a condition is a natural generalization of the well known result for ARX models. Although our algorithm is designed for perfect input/output data, our experiments also evaluate its performance as a function of the level of noise for different choices of the number of models and model orders. We also present an application to temporal video segmentation.     

Accuracy analysis of bias-eliminating least squares estimates for errors-in-variables systems

The bias-eliminating least squares (BELS) method is one of the consistent estimators for identifying dynamic errors-in-variables systems. In this paper, we investigate the accuracy properties of the BELS estimates. An explicit expression for the normalized asymptotic covariance matrix of the estimated parameters is derived and supported by some numerical examples.     

Convergence of the iterative algorithm for a general Hammerstein system identification

The convergence of the iterative identification algorithm for a general Hammerstein system has been an open problem for a long time. In this paper, it is shown that the convergence can be achieved by incorporating a regularization procedure on the nonlinearity in addition to a normalization step on the parameters.     

Generalized Yule-walker and two-stage identification algorithms for dual-rate systems

1 Introduction Presenting new identification methods and performance analysis of identification algorithms under weak conditions are everlasting themes and melodies of identification studies and are also my everlasting pursuits in life [1～4]. Multirate systems with different input and output sampling periods are very active in process industries [5～7], e.g. fermenta- tion processes [8], and petroleum production [9]. The stud- ies of multirate systems involve various areas of control, in- clu…     

Generalized Yule-walker and two-stage identification algorithms for dual-rate systems

In this paper, two approaches are developed for directly identifying single-rate models of dual-rate stochas- tic systems in which the input updating frequency is an integer multiple of the output sampling frequency. The first is the generalized Yule-Walker algorithm and the second is a two-stage algorithm based on the correlation technique. The basic idea is to directly identify the parameters of underlying single-rate models instead of the lifted models of dual-rate systems from the dual-rate input-output data, assuming that the measurement data are stationary and ergodic. An example is given.