Identification Of Multivariable Systems Research Articles

To the optimal identification of multivariate systems under perturbations of unknown covariances

We consider the problem of parameter and covariance estimation for multivariate stochastic systems described by regression models with special structure perturbations of unknown covariance. Sufficient conditions of uniformly optimal estimations are obtained for system parameters and covariances. The observation vector distribution family is factorized, and the full sufficient statistics is found under those conditions. Equations for uniformely optimal unbiased estimates of covariance parameters are obtained.

Modified subspace identification for long-range prediction model for inferential control

In a chemical plant containing a series of processing units, a model that can be used to forecast the behavior of downstream variables based on upstream measurements can be useful for feedforward and inferential control. With such, various upstream disturbances can be rejected well before they influence the downstream units. However, creating such a dynamic model can be a challenge. The conventional multivariable system identification approach, which minimizes single-step-ahead prediction error, can result in models leading to poor prediction and control in the described context. To alleviate this difficulty, a modification is proposed to the conventional subspace identification (SSID) method to emphasize the accuracy of k-step-ahead prediction, where k is a general integer. It is shown that the modified SSID method can be used in conjunction with the k-step prediction error minimization (PEM). Using illustrative examples of mixing units with a recycle loop and an industrial-relevant pulp digester, improvement from adopting the suggested modification is demonstrated.

Closed-loop identification of multivariable systems: With or without excitation of all references?

The accuracy of plant parameters estimated in closed-loop operation is investigated for a class of multivariable systems and for the situation where only some of the reference inputs are excited. This issue is important for at least two reasons: (i) there are control applications where it is preferable not to excite all references in order to avoid performance degradation, and (ii) it is not clear whether the results in the context of open-loop identification, where the existence of common parameters between the various transfer functions is a condition for improved accuracy with additional excitation, also hold in the closed-loop case. The paper examines the effect of the non-excitation of some reference inputs on the variance of the estimated parameters. The proposed expressions are valid for all conventional model structures used in prediction error identification. Although exciting all reference inputs is not necessary for identifiability, this work shows that, regardless of the parametrization, the excitation of all references never worsens and, in most cases, improves the accuracy of the parameter estimates. The analytical results developed in this work are illustrated by two simulation examples.

Multivariate System Identification for Cerebral Autoregulation

The effect of spontaneous beat-to-beat mean arterial blood pressure (ABP) fluctuations and breath-to-breath end-tidal carbon dioxide (PETCO2) and end-tidal oxygen (PETO2) fluctuations on beat-to-beat cerebral bloodflow velocity (CBFV) variations is studied using a multiple coherence function. Multiple coherence is a measure of the extent to which the output, CBFV, can be represented as a linear time invariant system of multiple input signals. Analysis of experimental measurements from 13 different healthy subjects reveal that, with additional inputs, PETCO2 and PETO2, the multiple coherence for frequencies <0.05 Hz is significantly higher than the corresponding values obtained for univariate coherence with a single input of ABP. The result illustrates that the low value of univariate coherence at small frequencies may be due to the effects of PETCO2 and PETO2 fluctuations on CBFV variability. Moreover, it is also found that the transfer function between ABP and CBFVtime series identified from previous univariate techniques at low frequencies can be modified by CO2 and O2 reactivity and no longer represents pressure autoregulation only. Multivariate system identification provides a technique of incorporating additional variability and recovering from this artifact. Finally, a physiologically based model and its linear transfer function are used as a simulation tool to investigate possible causes of low univariate coherence.

Comparison of input signals in subspace identification of multivariable ill-conditioned systems

Ill-conditioned processes often produce data of low quality for model identification in general, and for subspace identification in particular, because data vectors of different outputs are typically close to collinearity, being aligned in the “strong” direction. One of the solutions suggested in the literature is the use of appropriate input signals, usually called “rotated” inputs, which must excite sufficiently the process in the “weak” direction. In this paper open-loop (uncorrelated and rotated) random signals are compared against inputs generated in closed-loop operation, with the aim of finding the most appropriate ones to be used in multivariable subspace identification of ill-conditioned processes. Two multivariable ill-conditioned processes are investigated and as a result it is found that closed-loop identification gives superior models, both in the sense of lower error in the frequency response and in terms of higher performance when used to build a model predictive control system.

High-Purity Distillation

Distillation is one of the most common separation techniques in chemical manufacturing. This multi-input, multi-output staged separation process is strongly interactive, as determined by the singular value decomposition of a linear dynamic model of the system. Process dynamics associated with the low-gain direction are critical to the design of high-performance controllers for high-purity distillation but are difficult to estimate from conventional experimental test signals for identification. As a result, high-purity distillation columns are considered challenging cases for multivariable system identification and robust control system design. High-purity distillation is a challenging process application for system identification because of its nonlinear and strongly interactive dynamics. This article has described several constrained-optimization-based formulations for multisine input signal design that allow users to simultaneously specify the essential frequency- and time-domain properties of these signals. Because constraints are explicitly part of the design procedure, the approach is useful for accomplishing plant-friendly identification testing in the process industries. The problem formulations were evaluated for a highly nonlinear methanol-ethanol distillation column. Introducing directional sinusoids in the multisine signal, applying a closed-loop signal design, and minimizing an objective function based on Weyl's theorem enhanced the information content of the low-gain direction in the identification experiment.

ON THE PARAMETRIZATION OF MULTIVARIATE GARCH MODELS

This paper deals with issues of structure and parametrization of VECH models proposed in Bollerslev, Engle, and Wooldridge (1988) and Baba, Engle, Kraft, and Kroner (BEKK) models. Both general models and also restricted versions such as the widely used diagonal VECH (DVECH) and factor generalized autoregressive conditional heteroskedastic (F-GARCH) models are discussed. A simple algorithm is presented that checks whether a given VECH model may be cast as a BEKK model. It is shown that in the bivariate case BEKK models are as general as VECH models. In higher dimensional cases however, VECH models allow for more flexibility. In addition, a parametrization for a generic, i.e., open and dense, class of BEKK models is given, and the frequently cited parametrization by Engle and Kroner (1995, Econometric Theory 11, 122–150) is analyzed. Two shortcomings of the latter are pointed out. Finally, parametrizations for BEKK(p,q,K) models with K ≤ n, including DVECH, F-GARCH, and a generalization of the latter, are discussed.This research was supported by the project P17065 “Identification of multivariate dynamic systems with a focus on dimension reduction” of the Austrian Science Foundation (FWF).

An IV-QR Algorithm for Neuro-Fuzzy Multivariable Online Identification

In this paper, a new algorithm for neuro-fuzzy identification of multivariable discrete-time nonlinear dynamic systems, more specifically applied to consequent parameters estimation of the neuro-fuzzy inference system, is proposed based on a decomposed form as a set of coupled multiple input and single output (MISO) Takagi-Sugeno (TS) neuro-fuzzy networks. An on-line scheme is formulated for modeling a nonlinear autoregressive with exogenous input (NARX) recurrent neuro-fuzzy structure from input-output samples of a multivariable nonlinear dynamic system in a noisy environment. The adaptive weighted instrumental variable (WIV) algorithm by QR factorization based on the numerically robust orthogonal Householder transformation is developed to modify the consequent parameters of the TS multivariable neuro-fuzzy network

A multi-variate Hammerstein model for processes with input directionality

A new formulation of a block-structured model based on the Hammerstein operator is presented for the identification of multi-variate systems with input directionality. In contrast to the existing formulations for multi-variate Hammerstein models, the proposed structure offers the possibility to independently model the dynamic and nonlinear characteristics of the system and at the same time preserves the possibility to use the new efficient algorithms developed for the identification of single input Hammerstein models. Further, the formulation allows for a representation of arbitrary static nonlinear coupling of input variables with a considerably lower amount of parameters compared to existing formulations. The new model structure is applied to the identification of a fluid catalytic cracking (FCC) unit and significantly outperforms all previous multi-variate Hammerstein model structures by reducing the prediction error by over 50%.

FAST SYSTEM IDENTIFICATION AND MODEL REDUCTION SOLVERS

Efficient numerical techniques for multivariable system identification and model reduction are investigated. The techniques are implemented in the system identification and model reduction toolboxes based on the Fortran 77 Subroutine Library in Control Theory (SLICOT). Besides highly performant Fortran drivers and computational routines, these toolboxes provide MATLAB or Scilab interfaces, implementing several algorithmic approaches. Extensive numerical testing and comparisons with similar MATLAB tools show that the solvers in these toolboxes are reliable, efficient, and able to solve industrially relevant problems.

Multivariable system identification and generic model control of a laboratory scale packed distillation column

In this study, multivariable generic model control (MGMC) structure has been proposed to control of the packed distillation column. Top and bottom product temperatures of the packed distillation column were selected as controlled variables and reflux ratio and reboiler heat duty as manipulated variables, respectively. Regulator problem was considered for experimental control studies. Feed mole fraction and feed temperature were chosen as load variables. Two types black box models, which were linear and non-linear was used for generic model control to control of packed distillation. A multivariable non-linear autoregressive moving average with eXternal input and auto-recursive extensive models were used for the linear black box model. The system identification toolbox in MATLAB was applied to estimate of the discrete models. Non-linear black box model for MGMC has better performance over linear black box model in the case of load disturbances.

A Novel Linear Recurrent Neural Network for Multivariable System Identification

This paper proposes a novel linear recurrent neural network for multivariable system identification, namely a linerec neural network (LNN). Based on this network, the transfer function matrix model of a multivariable system can be identified directly according to its input and output data. In this way, LNNs differ from existing neural networks. An LNN is constructed based on the identification of prior knowledge in a system, and its weights have definite physical meaning. An LNN is equivalent to a linear equation set, and its training algorithm is based on Widrow-Hoff learning rules. In this paper, the theoretical foundation, structural algorithm and learning rules of LNNs are proposed and studied. To guarantee learning convergence, network training stability is analysed using discrete Lyapunov stability theory. Finally, simulation results show the feasibility of LNNs for multivariable system identification.

MIXED H2 AND H∞ INPUT DESIGN FOR MULTIVARIABLE SYSTEMS

In this contribution a new procedure for input design for identification of linear multivariable systems is proposed. The goal is to minimize the input power used in the system identification experiment. The quality constraint on the estimated model is formulated in ℋ∞. The input design problem is converted to linear matrix inequalities by a separation of graphs theorem. For illustration, the proposed method is applied on a chemical distillation column and the result shows that it is optimal to amplify the low gain direction of the plant.

CR-IDENT: A MATLAB TOOLBOX FOR MULTIVARIABLE CONTROL-RELEVANT SYSTEM IDENTIFICATION

This paper describes CR-IDENT, a Matlab-based toolbox that implements a comprehensive procedure for multivariable control-relevant system identification aimed primarily at process system applications. The toolbox consists of modules for multivariable input signal design (multisine and PRBS), frequency response estimation, and control-relevant frequency response curvefitting, leading to models whose end use is the design of high-performance control systems. An important component in the implementation of this design procedure is its reliance on a priori knowledge of the system of interest to design input signals meeting both theoretical and practical user requirements. Data from identification testing using these signals is the basis for the subsequent steps of frequency-response estimation and control-relevant parameter estimation, with the final result being a discrete-time state-space model that serves as the nominal model for Model Predictive Control. A high-purity distillation column example is presented to illustrate the benefits of the toolbox, from experiment design to closed-loop control.

High-performance numerical algorithms and software for structured total least squares

We present a software package for structured total least-squares approximation problems. The allowed structures in the data matrix are block-Toeplitz, block-Hankel, unstructured, and exact. Combination of blocks with these structures can be specified. The computational complexity of the algorithms is O ( m ) , where m is the sample size. We show simulation examples with different approximation problems. Application of the method for multivariable system identification is illustrated on examples from the database for identification of systems DAISY.

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.

Experimental Results on Stable Identification for Nonlinear Systems Using Neural Networks

This paper investigates the problem of stable identification of multivariable nonlinear systems using a multilayer feedforward neural network. Unlike most of the previous neural network identifiers, the proposed identifier is based on a nonlinear-in-parameters neural network (NLPNN). Therefore, it is applicable to systems with higher degrees of nonlinearities. Both parallel and series-parallel models are used with no a priori knowledge about the system dynamics. The method can be considered both as an online identifier that can be used as a basis for designing neural network controller as well as an off-line learning scheme for monitoring the system states. A novel approach is proposed for the weight updating mechanism based on the modification of the backpropagation algorithm. The stability of the overall system is shown using Lyapunov's direct method. To demonstrate the performance of the proposed algorithm, an experimental set-up consisting of a three-link macro-micro manipulator is considered. The proposed approach is applied to identify the dynamics of the experimental robot. Experimental results are given to show the effectiveness of the proposed learning scheme.

Gain scheduling control based on closed-loop system identification

This paper deals with system identification and gain scheduling control of multi-variable nonlinear systems. We propose a novel scheme where a linear approximation of the system model is obtained in an operating point; then, a Youla-Kucera (YJBK) parameter specifying the difference between the first and a second operating point is identified in closed-loop using system identification methods with open-loop properties. Next, a linear controller is designed for this linearised model, and gain scheduling control can subsequently be achieved by interpolating between each controller. In a constructive manner, we show how to use the YJBK parameterisation to achieve a smooth scheduling between the controllers. The approach is tested on a simple, but highly nonlinear model of a coal-fired power plant.

Efficient data processing for subspace-based multivariable system identification

Efficient, structure-exploiting techniques for input/output data processing in subspace-based multivariable system identification are investigated. The techniques are implemented in the system identification toolbox for discrete-time systems, SLIDENT, incorporated in the freely available Fortran 77 Subroutine Library in Control Theory (SLICOT). Besides drivers and computational routines, this toolbox provides MATLAB interfaces, implementing several algorithmic approaches. Extensive numerical testing and comparisons with similar MATLAB tools show that SLIDENT is reliable, efficient, and powerful enough to solve industrial identification problems.

High-performance numerical algorithms and software for subspace-based linear multivariable system identification

Basic algorithmic and numerical issues involved in subspace-based linear multivariable discrete-time system identification are described. A new identification toolbox—SLIDENT—has been developed and incorporated in the freely available Subroutine Library in Control Theory (SLICOT). Reliability, efficiency, and ability to solve industrial identification problems received a special consideration. Two algorithmic subspace-based approaches (MOESP and N4SID) and their combination, and both standard and fast techniques for data compression are provided. Structure exploiting algorithms and dedicated linear algebra tools enhance the computational efficiency and reliability. Extensive comparisons with the available computational tools based on subspace techniques show the better efficiency of the SLIDENT toolbox, at comparable numerical accuracy, and its capabilities to solve identification problems with many thousands of samples and hundreds of parameters.