Chung-Wen Chen

Frequency Domain State-Space System Identification

This paper presents an algorithm for identifying state-space models of linear systems from frequency response data. A matrix-fraction description of the transfer function is employed to curve-fit the frequency response data, using the least-squares method. The parameters of the matrix-fraction representation are then used to construct the Markov parameters of the system. Finally, state-space models are obtained through the Eigensystem Realization Algorithm using the Markov parameters. The main advantage of this approach is that the curve-fitting and the Markov-parameter-construction are linear problems which avoid the difficulties of non-linear optimization of other approaches. Another advantage is that it avoids windowing distortions associated with other frequency domain methods.

ESTIMATION OF KALMAN FILTER GAIN FROM OUTPUT RESIDUALS

This paper presents a procedure to estimate the Kalman filter gain from input-output measurement data with a given system model. The system model can be a finite element model or an experimental model from any identification method. The procedure consists of three basic steps. First, the stochastic portion related to the residuals of the response is computed. Second, the coefficients of a linear difference model for the stochastic portion are estimated by a least-squares solution that minimizes the filter residual. Third, the Kalman filter gain is computed from these model coefficients. Experimental results are presented to illustrate the usefulness of the developed procedure.

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.

Single-mode projection filters for modal parameter identification for flexible structures

Single-mode projection filters are developed for eigensystem parameter identification from both analytical results and test data. Explicit formulations of these projection filters are derived using the orthogonal matrices of the controllability and observability matrices in the general sense. A global minimum optimization algorithm is applied to update the filter parameters by using the interval analysis method. The updated modal parameters represent the characteristics of the test data. For illustration of this new approach, a numerical simulation for the MAST beam structure is shown by using a one-dimensional global optimization algorithm to identify modal frequencies and damping. Another numerical simulation of a ten-mode structure is also presented by using a two-dimensional global optimization algorithm to illustrate the feasibility of the new method. The projection filters are practical for parallel processing implementation.

Stable State-Space System Identification from Frequency Domain Data

Due to possible distortion contained in frequency-domain data, system identification methods based on data-matching alone do not guarantee stable models. This paper presents a method of identifying a stable state-space model of a linear system from its frequency response data. It is an improvement and extension of previous work. The method first identifies a matrix-fraction description of the transfer function matrix of the system from data. Then, through forming a multi-variable observable or controllable canonical form, the method separates and replaces the unstable sub-system by a stable sub-system having approximately the same frequency response. Finally, it calculates the Markov parameters of the resulting model and obtains a state-space model through the Eigensystem Realization Algorithm.

Identification of Stochastic System and Controller via Projection Filters

A method based on projection filters is presented for identifying an open-loop stochastic system with an existing feedback controller. The projection filters are derived from the relationship between the state-space model and the AutoRegressive with eXogeneous input (ARX) model including the system, Kalman filter and controller. Two ARX models are identified from the control input, closed-loop system response and feedback signal using least-squares method. Markov parameters of the open-loop system, Kalman filter and controller are then calculated from the coefficients of the identified ARX models. Finally, the state-space model of the open-loop stochastic system and the gain matrices for the Kalman filter and controller are realized. The method is validated by simulations and test data from an unstable large-angle magnetic suspension test facility.

Several recursive techniques for observer/Kalman filter system identification from data

This paper derives algorithms for identifying autoregressive models, with external input, of multi-input multi-output systems from data using a fast transversal filter or a least-squares lattice filter. The autoregressive models including external inputs are used to identify state-space models and the corresponding observer/Kalman filter gains of the system. The derivation is an extension of scalar autoregressive model approaches, modified to cope with multivariables, external inputs and an extra direct-influence term. Comparisons between the fast transversal filter, the least-squares lattice filter and the classical least-squares method are made in terms of complexity, computational cost and practical applications issues. A numerical example is included to illustrate the approach.

Identification of stochastic system and controller via projection filters

A method based on projection filters is presented for identifying an open-loop stochastic system with an existing feedback controller. The projection filters are derived from the relationship between the state-space model and the autoregressive with exogeneous input (ARX) model including the system, Kalman filter and controller. Two ARX models are identified from the control input, closed-loop system response and feedback signal using least-squares method. Markov parameters of the open-loop system, Kalman filter and controller are then calculated from the coefficients of the identified ARX models. Finally, the state-space model of the open-loop stochastic system and the gain matrices for the Kalman filter and controller are realized. The method is validated by simulations and test data from an unstable large-angle magnetic suspension test facility.

Single-mode projection filters for identification and state estimation of flexible structures

Single-mode projection filters are developed for eigensystem parameter identification and state estimation from both analytical results and test data. Explicit formulations of these projection filters are derived using the pseudoinverse matrices of the controllabilty and observability matrices in the general sense. A global minimum optimization algorithm is developed to update the filter parameters by using the interval analysis method. Modal parameters can be identified and updated in the global sense within a specified region of parameters by passing the experimental data through the projection filters. For illustration of this new approach, a numerical example is shown by using a one-dimensional global optimization algorithm to estimate modal frequencies and damping.

State Estimation Under Unknown Noises — A Least-Squares Approach

A simple filter for state estimation of linear systems under unknown noises is developed. Through state space model, the current state of the system is linked to previous measurement data and can be estimated by using least-squares techniques. Due to the unknown process noise, past data are regarded as of decaying importance in determining the current state and a forgeting factor is employed in the recursive least-squares to function fading memory. The relations between the fading memory filter and the Kaiman filter are discussed. A numerical example is given to illustrate the feasibility of the approach.