Identification Of Linear Stochastic Systems Research Articles

Generalized adaptive comb filters/smoothers and their application to the identification of quasi-periodically varying systems and signals

The problem of both causal and noncausal identification of linear stochastic systems with quasi-harmonically varying parameters is considered. The quasi-harmonic description allows one to model nonsinusoidal quasi-periodic parameter changes. The proposed identification algorithms are called generalized adaptive comb filters/smoothers because in the special signal case they reduce down to adaptive comb algorithms used to enhance or suppress nonstationary harmonic signals embedded in noise. The paper presents a thorough statistical analysis of generalized adaptive comb algorithms, and demonstrates their statistical efficiency in the case where the fundamental frequency of parameter changes varies slowly with time according to the integrated random-walk model.

Bayesian identification of stochastic linear systems with observations at multiple scales

A Bayesian system identification approach to modelling stochastic linear dynamical systems involving multiple scales is proposed, where multiscale means that the output of the system is observed at one (coarse) resolution whilst the input of the system can only be observed at another (fine) resolution. The proposed method identifies linear models at different levels of resolution, where the link between the two resolutions is realised via a non-overlapping averaging process. The averaged data at the coarse level of resolution is assumed to be a set of observations from an implied process so that the implied process and the output of the system result in an errors-in-variables model at the coarse level of resolution. By using a Bayesian inference and Markov chain Monte Carlo method, such a modelling framework results in different dynamical models at different levels of resolution at the same time. The new method is also shown to have the ability to combine information across different levels of resolution. Simulation examples are provided to show the efficiency of the new method. Furthermore, an application to the analysis of the relativistic electron intensity at the geosynchronous orbit is also included.

OPTIMAL INPUT DESIGN FOR LINEAR STOCHASTIC MODEL IDENTIFICATION FROM VIEWPOINT OF MULTI-OBJECTIVE OPTIMIZATION

System identification deals with the problems of building mathematical models of the systems based on observed input/output data from the systems. Optimal experimental design for system identification to extract the maximum information about the system has been extensively investigated, such as optimal design of input, sampling intervals, pre-filters, etc. In system identification, there are model structure determination and parameter estimation steps, and the optimality criteria for both objects are sometimes conflicting. This leads the necessity of the compromises in the design, i.e., the tradeoffs between the performances in these two steps should be introduced. We investigate in this paper the optimal input design for identification of linear stochastic systems from the viewpoint of multi-objective optimization problem. The Pareto-optimal set of inputs is derived and how it is used in system identification is discussed.

Risk sensitive identification of linear stochastic systems

Risk-sensitive identification of AR-processes was first considered in [12]. The purpose of this paper is to extend this original approach to ARMA-processes and even to multi-variable linear stochastic systems. We provide a new definition of a risk-sensitive identification criterion. For this we first consider a recursive identification procedure which is parameterized by a weight-matrix K acting on the stochastic gradient. Using the asymptotic theory of recursive estimation a suitably scaled version of the error process will be approximated by a stationary Gaussian process, see Chapter 4.5, Part II of [1]. The new risk sensitive criterion will be defined in terms of this associated stationary Gaussian process in a familiar manner via an exponential-quadratic cost. The main result of the paper is the minimization of the proposed new criterion with respect to the weight-matrix K over a feasible set EK, see (22), where the cost function is known to be finite, Theorem 6.1. This results will then be extended to the case when minimization over a feasible set E°K is considered, see (26), on the complement of which the cost function is known to be infinite, Theorem 6.1. The starting point of our analysis is an expression of the cost function given in LEQG-theory, in particular a result of [10]. A new expression for the cost function will be also given, using stochastic realization theory, as the mutual information rate between two stochastic processes.

On least‐squares identification of stochastic linear systems with noisy input–output data

On least‐squares identification of stochastic linear systems with noisy input–output data

Comments on "On a least-squares-based algorithm for identification of stochastic linear systems"

The bias-eliminated least squares estimator proposed in the paper by Zheng (see ibid., vol.46, p.1631-8, 1998) is shown to be identical to a simple instrumental variable estimator, using delayed input values as instruments.

On a least-squares-based algorithm for identification of stochastic linear systems

A new form of bias-eliminated least-squares (BELS) algorithm is developed to identify transfer function parameters of a linear time-invariant system, irrespective of noise dynamics. Unlike the BELS estimator previously presented, the main feature with the developed algorithm is that the transfer function parameters are consistently estimated in such a direct way that there is no need to prefilter observed data or to deal with a high-order augmented system. This greatly simplifies implementation of the BELS-based algorithms and reduces numerical efforts, whereas a desirable estimation accuracy can still be achieved. Two simulation examples are presented that clearly illustrate the good performances of the developed algorithm, including its superiority over one type of simple instrumental variable method.

Indirect identification of linear stochastic systems with known feedback dynamics

An algorithm is presented for identifying a state-space model of linear stochastic systems operating under known feedback controller. In this algorithm, only the reference input and output of closed-loop data are required. No feedback signal needs to be recorded. The overall closed-loop system dynamics is first identified. Then a recursive formulation is derived to compute the open-loop plant dynamics from the identified closed-loop system dynamics and known feedback controller dynamics. The controller can be a dynamic or constant-gain full-state feedback controller. Numerical simulations and test data of a highly unstable large-gap magnetic suspension system are presented to demonstrate the feasibility of this indirect identification method.

Identification of linear stochastic systems through projection filters

A novel method is presented for identifying a state-space model and a state estimator for linear stochastic systems from input and output data. The method is primarily based on the relationship between the state-space model and the finite difference model of linear stochastic systems derived through projection filters. It is proved that least-squares identification of a finite difference model converges to the model derived from the projection filters. System pulse response samples are computed from the coefficients of the finite difference model. In estimating the corresponding state estimator gain, a z-domain method is used. First the deterministic component of the output is subtracted out, and then the state estimator gain is obtained by whitening the remaining signal. An experimental example is used to illustrate the feasibility of the method. YSTEM identification, sometimes also called system modeling, deals with the problem of building a mathematical model for a dynamic system based on its input/output data. This technique is important in many disciplines such as economics, communication, and system dynamics and control.1 The mathematical model allows researchers to understand more about the properties of the system, so that they can explain, predict, or control the behaviors of the system. Recently, a method has been introduced in Refs. 2 and 3 to iden- tify a state-space model from a finite difference model. The differ- ence model, called autoregressive with exogeneous input (ARX), is derived through Kalman filter theories. However, the method re- quires to use an ARX model of large order, which causes intensive computation in the embedded least-squares operation. In Ref. 4 a method is derived to obtain a state-space model from input/output data using the notion of state observers. This approach can use an ARX model with an order much smaller than that derived through the Kalman filter, but the derivation is based on a deterministic ap- proach. In Ref. 5, it has been proved that, as the order of the ARX model increase to infinity, the observer identification converges to the Kalman filter identification. However, for a stochastic system and an ARX model of a small order, to what the least-squares iden- tification of the ARX model will converge in a stochastic sense is not clear. This paper addresses the above-mentioned problems using a stochastic approach. The approach is primarily based on the re- lationship between the state-space model and the finite difference model via the projection filter.3 First, an ARX model is chosen, and then the ordinary least squares is used to estimate the coefficient matrices. Based on the relationship between the projection filter and the state-space model matrices, the system pulse response samples (i.e., the system Markov parameters) can be calculated from the co- efficients of the identified ARX model. The eigensystem realization algorithm (ERA)6 is used to decompose the Markov parameters into a state-space model. In contrast to the time-domain approaches used in Refs. 2 and 5, a different method is developed in this paper using a z-domain approach to compute the state estimator gain. After identifying a state-space model, the deterministic part of the output is subtracted out. The remaining signal represents the stochastic part. A moving- average (MA) model is then introduced to describe the remaining signal. The MA model is computed by identifying the correspond- ing autoregressive (AR) model first and then inverting it. From the identified MA model, the state estimator gain is then calculated. Finally, identification of a 10-bay structure is used to illustrate the feasibility of the approach.

Identification of linear stochastic systems based on partialinformation

In this paper, we consider an identification problem for a system of partially observed linear stochastic differential equations. We present a result whereby one can determine all the system parameters including the covariance matrices of the noise processes. We formulate the original identification problem as a deterministic control problem and prove the equivalence of the two problems. The method of simulated annealing is used to develop a computational algorithm for identifying the unknown parameters from the available observation. The procedure is then illustrated by some examples.

Consistent identification of stochastic linear systems with noisy input-output data

A novel criterion is introduced for parametric errors-in-variables identification of stochastic linear systems excited by non-Gaussian inputs. The new criterion is (at least theoretically) insensitive to a class of input-output disturbances because it implicitly involves higher- than second-order cumulant statistics. In addition, it is shown to be equivalent to the conventional Mean-Squared Error (MSE) as if the latter was computed in the ideal case of noise-free input-output data. The sampled version of the criterion converges to the novel MSE and guarantees strongly consistent parameter estimators. The asymptotic behavior of the resulting parameter estimators is analyzed and guidelines for minimum variance experiments are discussed briefly. Informative enough input signals and persistent of excitation conditions are specified. Computatonally attractive Recursive-Least-Squares variants are also developed for on-line implementation of ARMA modeling, and their potential is illustrated by applying them to time-delay estimation in low SNR environment. The performance of the proposed algorithms and comparisons with conventional methods are corroborated using simulated data.

Robust identification of stochastic linear systems with correlated noise

The normal least-squares (LS) approach is one of the most simple and powerful identification methods widely used. Unfortunately, direct implementation of the LS method can give rise to biased or nonconsistent estimates of system parameters in the presence of correlated disturbances. In this paper, an up-to-date LS based identification approach is introduced to obtain consistent parameter estimates for stochastic systems subject to correlated noise. A designed filter is inserted into the identified system so that the resulting system has some known zeros which can, based on asymptotic analysis, be used for eliminating the coloured-noise-induced bias in the LS estimators. It is shown that the proposed identification method can not only produce consistent estimates, but be a very feasible, robust identification technique in practical applications as well. The theoretical analysis is verified through the Monte-Carlo stochastic simulation studies.

System Identification of Multivariate Systems With Feedback

System identification of linear stochastic systems has been the concern of many scientists and engineers. The same system can be described in many ways and consequently many different models have been proposed. However, the complexity of the systems and computational difficulty, especially under the presence of feedback and external disturbances, have rendered identification a difficult task. This paper deals with a general modeling procedure that can be used for identification of feedback systems. First, a brief discussion on the joint input-output process models is given, followed by the discussion on the canonical representation of the model. A simplified model is derived from the general vector model. This will lead to the validity of using the Modified Autoregressive Moving Average Vector (MARMAV) model in the identification of the general multivariate system with open-loop and closed-loop dynamics. Next, an estimation procedure is explained. The estimation is pursued to reach a maximum likelihood model by nonlinear iterative calculations. Since initial values are required to start the procedure which are critical to the estimation, two methods for the initial guess value estimation are provided in the appendices.

A method for adaptive estimation of ARMA processes

When the noise process in adaptive identification of linear stochastic systems is correlated, and can be represented by a moving average model, extended least squares algorithms are commonly used, and converge under a strictly positive real (SPR) condition on the noise model. In this paper, we present an adaptive algorithm for the estimation of autoregressive moving average (ARMA) processes, and show that it is convergent without any SPR condition, and has a convergence rate of O({ loglog t)/t} 1 2 ) .

Identification of linear stochastic systems via second- and fourth-order cumulant matching

The identification problem for time-invariant single-input single-output linear stochastic systems driven by non-Gaussian white noise is considered. The system is not restricted to be minimum phase, and it is allowed to contain all-pass components. A least-squares criterion that involves matching the second- and the fourth-order cumulant functions of the noisy observations is proposed. Knowledge of the probability distribution of the driving noise is not required. An order determination criterion that is a modification of the Akaike information criterion is also proposed. Strong consistency of the proposed estimator is proved under certain sufficient conditions. Simulation results are presented to illustrate the method.

Identification of stochastic linear systems in presence of input noise

Most identification methods rely on the assumption that the input is known exactly. However, when collecting data under an identification experiment it may not be possible to avoid noise when measuring the input signal. In the paper some different ways to identify systems from noisy data are discussed. Sufficient conditions for identifiability are given. Also accuracy properties and the computational requirements are discussed. A promising approach is to treat the measured input and output signals as outputs of a multivariable stochastic system. If a prediction error method is applied using this approach the system will be identifiable under mild conditions.

Optimal policies for identification of stochastic linear systems

The problem of designing closed-loop policies for identification of multiinput-multioutput linear discrete-time systems with random time-varying parameters is considered in this paper using a Bayesian approach. A sensitivity index gives a measure of performance for the closed-loop laws. The computation of the optimal laws is shown to be nontrivial, an exercise in stochastic control, but open-loop, affine, and open-loop feedback optimal inputs are shown to yield tractable problems. Numerical examples are given. For time-invariant systems, the criterion considered is shown to be related to the trace of the information matrix associated with the system.

Discussion of the Paper by Dr Brown, Professor Durbin and Mr Evans

Discussion of the Paper by Dr Brown, Professor Durbin and Mr Evans

Maximum likelihood identification of stochastic linear systems

The maximum likelihood estimation of the coefficients of multiple output linear dynamical systems and the noise correlations from the noisy measurements of input and output are discussed. Conditions are derived under which the estimates converge to their true values as the number of measurements tend to infinity. The computational methods are illustrated by several numerical examples.