On the frequency domain accuracy of closed-loop estimates

It has been argued that the frequency domain accuracy of high model-order estimates obtained on the basis of closed-loop data is largely invariant to whether direct or indirect approaches are used. The analysis underlying this conclusion has employed variance expressions that are asymptotic both in the data length and the model order, and hence are approximations when either of these are finite. However, recent work has provided variance expressions that are exact for finite (possibly low) model order, and hence can potentially deliver more accurate quantification of estimation accuracy. This paper, and a companion one, revisits the study of identification from closed-loop data in light of these new quantifications and establishes that, under certain assumptions, there can be significant differences in the accuracy of frequency response estimates. These discrepencies are established here and in the companion paper to be dependent on what type of direct, indirect or joint input-output identification strategy is pursued.     

Closed-loop identification: a two step approach

The accuracy aspects of identification (with respect to both variance and bias of estimates) and the role of filtering in closed-loop identification is discussed in this paper. It is shown that the key difference between closed-loop and open-loop identification is the existence of the sensitivity function. A closed-loop identification algorithm which asymptotically yields the same expressions as open-loop identification, in both variance and bias errors, is proposed. The proposed algorithm is evaluated by simulated examples as well as experiments performed on a computer-interfaced pilot-scale process.     

Identification for control: Optimal input intended to identify a minimum variance controller

It is well known that the quality of the parameters identified during an identification experiment depends on the applied excitation signal. Prediction error identification using full order parametric models delivers an ellipsoidal region in which the true parameters lie with some prescribed probability level. This ellipsoidal region is determined by the covariance matrix of the parameters. Input design strategies aim at the minimization of some measure of this covariance matrix. We show that it is possible to optimize the input in an identification experiment with respect to a performance cost function of a closed-loop system involving explicitly the dependence of the designed controller on the identified model. In the present contribution we focus on finding the optimal input for the estimation of the parameters of a minimum variance controller, without the intermediate step of first minimizing some measure of the model parameter accuracy. We do this in conjunction with using covariance formulas which are not asymptotic in the model order, which is rather new in the domain of optimal input design. The identification procedure is performed in closed-loop. Besides optimizing the input power spectrum for the identification experiment, we also address the question of optimality of the controller. It is a wide belief that the minimum variance controller should be the optimal choice, since we perform an experiment for designing a minimum variance controller. However, we show that this may not always be the case, but rather depends on the model structure.     

Comparison of the closed-loop identification methods in terms of the bias distribution

Several schemes for plant model identification in closed-loop operation including classical direct method, two-step identification and closed-loop output error algorithms are considered. These methods are analyzed and compared in terms of the bias distribution of the estimates for the case that the noise model is estimated as well as the case that a fixed model of noise is considered (output error structure). The problems concerning the filtered direct method which is often used in the iterative identification and control scheme are mentioned. It is shown that these problems may be solved by the closed-loop output error identification method.     

Conditions when minimum variance control is the optimal experiment for identifying a minimum variance controller

It is well known that if we intend to use a minimum variance control strategy, which is designed based on a model obtained from an identification experiment, the best experiment which can be performed on the system to determine such a model (subject to output power constraints, or for some specific model structures) is to use the true minimum variance controller. This result has been derived under several circumstances, first using asymptotic (in model order) variance expressions but also more recently for ARMAX models of finite order. In this paper we re-approach this problem using a recently developed expression for the variance of parametric frequency function estimates. This allows a geometric analysis of the problem and the generalization of the aforementioned finite model order ARMAX results to general linear model structures.     

Frequency domain identification with generalized orthonormal basisfunctions

A method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain, To this end, use is made of the empirical transfer function estimate (ETFE), identified from time-domain data. As a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions, being generalizations of the classical pulse and Laguerre and Kautz types of bases. An asymptotic analysis of the estimated models is provided and conditions for consistency are formulated. Explicit and transparent bias and variance expressions are established, the latter ones also valid in a situation of undermodeling     

A virtual closed loop method for closed loop identification

Indirect methods for the identification of linear plant models on the basis of closed loop data are based on the use of (reconstructed) input signals that are uncorrelated with the noise. This generally requires exact (linear) controller knowledge. On the other hand, direct identification requires exact plant and noise modelling (system in the model set) in order to achieve accurate results, although the controller can be non-linear. In this paper, a generalized approach to closed loop identification is presented that includes both methods as special cases and which allows novel combined methods to be generated. Besides providing robustness with respect to inexact controller knowledge, the method does not rely on linearity of the controller nor on exact noise modelling. The generalization is obtained by balancing input-noise decorrelation against noise whitening in a user-chosen flexible fashion. To this end, a user-chosen virtual controller is used to parametrize the plant model, thereby generalizing the dual-Youla method to cases where knowledge of the controller is inexact. Asymptotic bias and variance results are presented for the method. Also, the benefits of the approach are demonstrated via simulation studies.     

Subspace identification with non-steady Kalman filter parameterization

Most existing subspace identification methods use steady-state Kalman filter (SKF) in parameterization, hence, infinite data horizons are implicitly assumed to allow the Kalman gain to reach steady state. However, using infinite horizons requires collecting infinite data which is unrealistic in practice. In this paper, a subspace framework with non-steady state Kalman filter (NKF) parameterization is established to provide exact parameterization for finite data horizon identification problems. Based on this we propose a novel subspace identification method with NKF parameterization which can handle closed-loop data and avoid assumption on infinite horizons. It is shown that with finite data, the proposed parameterization method provides more accurate and consistent solutions than existing SKF based methods. The paper also reveals why it is often beneficial in practice to estimate a bank of ARX models over a single ARX model.     

Finite sample properties of indirect nonparametric closed-loop identification

This paper presents new results on the properties of indirect nonparametric estimation using closed-loop data. Specific results developed include finite sample bias and variance. We show that previous asymptotic results hold only when the signal-to-noise ratio is large. We develop an expression which holds generally and which departs significantly from the known asymptotic results. Simulations are presented which substantiate the validity of the general expression.     

Variance-error quantification for identified poles and zeros

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one, but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box–Jenkins models.     

Identification of probabilistic system uncertainty regions byexplicit evaluation of bias and variance errors

A procedure is developed for identification of probabilistic system uncertainty regions for a linear time-invariant system with unknown dynamics, on the basis of time sequences of input and output data. The classical framework is handled in which the system output is contaminated by a realization of a stationary stochastic process. Given minor and verifiable prior information on the system and the noise process, frequency response, pulse response, and step response confidence regions are constructed by explicitly evaluating the bias and variance errors of a linear regression estimate. In the model parametrizations, use is made of general forms of basis functions. Conservatism of the uncertainty regions is limited by focusing on direct computational solutions rather than on closed-form expressions. Using an instrumental variable method for identification, the procedure is suitable also for input-output data obtained from closed-loop experiments     

On the assessment of multivariable controllers using closed loop data. Part I: Identification of system models

Controller performance assessment of SISO and MIMO systems requires effective and systematic identification of the associated system models based on closed-loop data. In this work, a new methodology for the identification of the process, controller and disturbance models is presented for the purpose of enabling the evaluation of the performance of MIMO control systems. The methodology is based on subspace identification algorithms for the identification of the controller, process and disturbance models from closed-loop data. However, identification of the process model is enhanced by the estimation of the associated interactor matrix via the Variable Regression Estimation technique, the existence of which is mathematically proved. The proposed identification methodology is applied to two 2 × 2 systems utilizing both step-response and PRBS closed-loop data.     

Identification of dynamic models in complex networks with prediction error methods—Basic methods for consistent module estimates

The problem of identifying dynamical models on the basis of measurement data is usually considered in a classical open-loop or closed-loop setting. In this paper, this problem is generalized to dynamical systems that operate in a complex interconnection structure and the objective is to consistently identify the dynamics of a particular module in the network. For a known interconnection structure it is shown that the classical prediction error methods for closed-loop identification can be generalized to provide consistent model estimates, under specified experimental circumstances. Two classes of methods considered in this paper are the direct method and the joint-IO method that rely on consistent noise models, and indirect methods that rely on external excitation signals like two-stage and IV methods. Graph theoretical tools are presented to verify the topological conditions under which the several methods lead to consistent module estimates.     

Persistent excitation condition within the dual control framework

Model Predictive Control framework is currently used in many different fields of expertise. The inherent part and very often also the main bottleneck is the model of a process used for the computation of predictions.Due to many reasons e.g. ageing, from time to time there exists a need to adjust/re-identify (if there was already some kind of a model-based controller) or to construct a brand new model (in other cases). Frequently, the process generating the data is under some kind of control, imposing thus problems when classical open loop identification methods are considered. The need for models identified from the data gathered in a closed-loop fashion and a request for possible re-identification of the model parameters lead to the emerge of dual control where the problems of control and system identification are addressed simultaneously.In this paper, we present a new algorithm based on the persistent excitation condition when the minimal eigenvalue of the information matrix is maximized in order to have sufficiently exciting optimal control signal satisfying the control requirements.     

Closed-loop identification using direct approach and high order ARX/GOBF-ARX models

Model accuracy plays a key role in the performance of advanced, model predictive control algorithms. Model fidelity is usually affected by routine operating condition changes, which necessitate reidentification. From several theoretical and practical considerations, it is recommended that such re-identification be performed under closed-loop conditions. The direct approach for closed-loop identification, owing to its simplicity, is better suited for MPC. In order to yield unbiased and consistent parameter estimates, however, this approach requires the noise model to be sufficiently parameterized. Towards this objective, high order ARX models are the most suitable candidates from the viewpoint of ease of parameter estimation. For multivariable systems, however, the identification of high order ARX models would require longer experiments to be performed. This being undesirable from a practical viewpoint, there is a need for a parsimonious parameterization that would retain the benefits of high order ARX models. In this work, we propose to use generalized orthonormal basis filters (GOBFs) to achieve this parsimonous parameterization. Further, we propose an approach to obtain reduced order models by emphasizing important frequencies so as to suitably shape the bias. We also show that the choice of the GOBF parameterization has another important merit, viz. their ability to perform well even with minimal perturbation data or short experiment times. The efficacy of the proposed approach is demonstrated via simulations on the benchmark Shell Control Problem and a laboratory quadruple tank setup.     

System identification using Kautz models

In this paper, the problem of approximating a linear time-invariant stable system by a finite weighted sum of given exponentials is considered. System identification schemes using Laguerre models are extended and generalized to Kautz models, which correspond to representations using several different possible complex exponentials. In particular, linear regression methods to estimate this sort of model from measured data are analyzed. The advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters. The subsequent analysis is based on the result that the corresponding linear regression normal equations have a block Toeplitz structure. Several results on transfer function estimation are extended to discrete Kautz models, for example, asymptotic frequency domain variance expressions     

Multi-output process identification

In model based control of multivariate processes, it has been common practice to identify a multi-input single-output (MISO) model for each output separately and then combine the individual models into a final MIMO model. If models for all outputs are independently parameterized then this approach is optimal. However, if there are common or correlated parameters among models for different output variables and/or correlated noise, then performing identification on all outputs simultaneously can lead to better and more robust models. In this paper, theoretical justifications for using multi-output identification for a multivariate process are presented and the potential benefits from using them are investigated via simulations on two process examples: a quality control example and an extractive distillation column. The identification of both the parsimonious transfer function models using multivariate prediction error methods, and of non-parsimonious finite impulse response (FIR) models using multivariate statistical regression methods such as partial least squares (PLS2), canonical correlation regression (CCR) and reduced rank regression (RRR) are considered. The multi-output identification results are compared to traditional single-output identification from several points of view: best predictions, closeness of the model to the true process, the precision of the identified models in frequency domain, stability robustness of the resulting model based control system, and multivariate control performance. The multi-output identification methods are shown to be superior to the single-output methods on the basis of almost all the criteria. Improvements in the prediction of individual outputs and in the closeness of the model to the true process are only marginal. The major benefits are in the stability and performance robustness of controllers based on the identified models. In this sense the multi-output identification methods are more ‘control relevant’.     

Instrumental variable methods for closed-loop system identification

In this paper, several instrumental variable (IV) and instrumental variable-related methods for closed-loop system identification are considered and set in an extended IV framework. Extended IV methods require the appropriate choice of particular design variables, as the number and type of instrumental signals, data prefiltering and the choice of an appropriate norm of the extended IV-criterion. The optimal IV estimator achieves minimum variance, but requires the exact knowledge of the noise model. For the closed-loop situation several IV methods are put in an extended IV framework and characterized by different choices of design variables. Their variance properties are considered and illustrated with a simulation example.     

Finite sample properties of system identification methods

In this paper we study the quality of system identification models obtained using the standard quadratic prediction error criterion for a general linear model class. The main feature of our results is that they hold true for a finite data sample and they are not asymptotic. The main theorems bound the difference between the expected value of the identification criterion evaluated at the estimated parameters and at the optimal parameters. The bound depends naturally on the model and system order, the pole locations, and the noise variance, and it shows that although these variables often do not enter in asymptotic convergence results, they do play an important role when the data sample is finite.