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.     

For model-based control design, closed-loop identification gives better performance

We compare open loop versus closed loop identification when the identified model is used for control design, and when the system itself belongs to the model class, so that only variance errors are relevant. Our measure of controller performance (which is used as our design criterion for identification) is the variance of the error between the output of the ideal closed loop system (with the ideal controller) and that of the actual closed loop system (with the controller computed from the identified model). Under those conditions, we show that, when the controller is a smooth function of the input-output dynamics and the disturbance spectrum, the best controller performance is achieved by performing the identification in closed loop with an operating controller that we characterize. For minimum variance and model reference control design criteria, we show that this ‘optimal operating controller for identification’ is the ideal controller. This then leads to a suboptimal but feasible iterative scheme.     

Subspace identification of Bilinear and LPV systems for open- and closed-loop data

In this paper we present a novel algorithm to identify LPV systems with affine parameter dependence operating under open- and closed-loop conditions. A factorization is introduced which makes it possible to form a predictor that predicts the output, which is based on past inputs, outputs, and scheduling data. The predictor contains the LPV equivalent of the Markov parameters. Using this predictor, ideas from closed-loop LTI identification are developed to estimate the state sequence from which the LPV system matrices can be constructed. A numerically efficient implementation is presented using the kernel method. It turns out that if structure is present in the scheduling sequence the computational complexity reduces even more.     

A practical global multi-stage method for fully automated closed-loop identification of industrial processes

A method for automated closed-loop identification is presented. The method is focused on practical deployment and has evolved based on academic developments, personal experience in industrial applications and feedback from seasoned practitioners who have a wealth of experience in implementing multivariable predictive control (MPC) and optimization projects. In essence, the basic idea here is that no single method will be adequate for the range of conditions encountered in the target industries. Hence this approach utilizes a family of prediction error model derived structures and consequently a family of model orders for each structure. These families are then searched for the most effective model for the given application. Key in this search is the definition of metrics not exclusively dependent on asymptotic theory. Extensive plant data are used to illustrate important features of the approach and overall performance.     

The nature of data pre-filters in MPC relevant identification—open- and closed-loop issues

In this paper a model predictive control relevant identification (MRI) method is applied to a general class of linear PEM models and the effect of bias distribution on the multistep ahead predictions is studied. Good multistep ahead predictions are essential for model predictive controllers. Therefore, it is important to distribute the bias in such a way that it is compatible with the predictive control objective. This paper deals with the impact of MRI methods on the bias distribution and its effect on control loop performance.     

Proportional stabilization and closed-loop identification of an unstable fractional order process

This paper deals with proportional stabilization and closed-loop step response identification of the fractional order counterparts of the unstable first order plus dead time (FOPDT) processes. At first, the necessary and sufficient condition for stabilizability of such processes by proportional controllers is found. Then, by assuming that a process of this kind has been stabilized by a proportional controller and the step response data of the closed-loop system is available, an algorithm is proposed for estimating the order and the parameters of an unstable fractional order model by using the mentioned data.     

Consistency analysis of some closed-loop subspace identification methods

We study statistical consistency of two recently proposed subspace identification algorithms for closed-loop systems. These algorithms may be seen as implementations of an abstract state-space construction procedure described by the authors in previous work on stochastic realization of closed-loop systems. A detailed error analysis is undertaken which shows that both algorithms are biased due to an unavoidable mishandling of initial conditions which occurs in closed-loop identification. Instability of the open loop system may also be a cause of trouble.     

Quantification of frequency domain error bounds with guaranteed confidence level in prediction error identification

This paper considers prediction error identification of linearly parametrized models in the situation where the system is in the model set. For such situation it is easy to construct a confidence ellipsoid in parameter space in which the true parameter lies with an a priori fixed probability level, α. Surprisingly perhaps, the construction of a corresponding uncertainty set in the frequency domain, to which the true system belongs with probability α, is still an open problem. We show in this paper how to construct such frequency domain uncertainty set with a probability level of at least α.     

The variation of non-parametric estimates in closed-loop

It is known that non-parametric transfer function estimates in closed-loop often have infinite variance. We characterise the probability density function of such estimates under the assumption that the corresponding closed-loop system estimate has complex normal distribution in the frequency domain. The probability density function can be described as a horseshoe encircling the inverse of the controller, with a global maximum on the line between the true value and the inverse of the controller. The expected value of the absolute value of such estimates is finite, and we propose it as a measure of variation. We also derive and discuss new expressions for the variance when an exclusion zone is introduced around the singularity.     

set membership identification: A survey

Robustness had become in past years a central issue in system and control theory, focusing the attention of researchers from the study of a single model to the investigation of a set of models, described by a set of perturbations of a “nominal” model. Such a set, often indicated as an uncertainty model set or model set for short, has to be suitably constructed to describe the inherent uncertainty about the system under consideration and to be used for analysis and design purposes. H_∞ identification methods deliver uncertainty model sets in a suitable form to be used by well-established robust design techniques, based on H_∞ or μ optimization methods. The literature on H_∞ identification is now very extensive. In this paper, some of the most relevant contributions related to assumption validation, evaluation of bounds on unmodeled dynamics, convergence analysis and optimality properties of linear, two-stage and interpolatory algorithms are surveyed from a deterministic point of view.     

Impact of new identification and tracking technologies on a distribution center

Automatic identification is a broad term given to a host of technologies that are used to help machines identify objects. Automatic identification is often coupled with automatic data capturing technology as companies with interests in logistics want to identify items, capture information about them and somehow enter the data into a computer without having employees type it in. The aim of automatic identification systems is to increase efficiency, reduce data entry errors, and free up staff to perform more value-added functions.     

Model validation for control and controller validation in a prediction error identification framework—Part II: illustrations

In this paper, we illustrate our new results on model validation for control and controller validation in a prediction error identification framework, developed in a companion paper (Gevers et al., Automatica (2003) 39(3) pii: S005-1098(02)00234-0), through two realistic simulation examples, covering widely different control design applications. The first is the control of a flexible mechanical system (the Landau benchmark example) with a tracking objective, the second is the control of a ferrosilicon production process with a disturbance rejection objective.     

Control relevant identification for robust optimal control

This paper presents an approach to designing the input signal for an identification experiment, in which the process model estimate is to be used to formulate and solve for a robust (in a worst case sense) optimal controller. The input signal is designed to contain the information that is relevant for the end use of the model, that is for control purposes. The proposed approach uses sensitivity analysis to determine the input signal frequencies that are important with respect to a certain measure of achievable controller performance in conjunction with a frequency sampling filter model of the process. Based on the sensitivity analysis, an iterative experimental design methodology is suggested.     

A theoretical analysis of recursive identification methods

In this paper five different recursive identification methods will be analyzed and compared, namely recursive versions of the least squares method, the instrumental variable method, the generalized least squares method, the extended least squares method and the maximum likelihood method. They are shown to be similar in structure and need of computer storage and time. Making use of recently developed theory for asymptotic analysis of recursive stochastic algorithms, these methods are examined from a theoretical viewpoint. Possible convergence points and their global and local convergence properties are studied. The theoretical analysis is illustrated and supplemented by simulations.     

Model uncertainty in frequency response based system identification

Determination of plant nominal model error bound is one of the most essential topics in robust control oriented identification. The objective of this paper is to investigate the structure of plant model uncertainties when a set of corrupted plant frequency response samples are supplied. It is shown that the plant model uncertainties, which result from the “partialness” and “corruptedness” of the provided plant information, can be represented by structure-fixed, norm-bounded but uncertain matrix-valued functions that perturb a plant nominal model through a linear fractional transformation. These results are very similar to those when plant time domain identification experiment data have been provided.     

Relations between uncertainty structures in identification for robust control

Various techniques of system identification exist that provide a nominal model and an uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. It is shown that an amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and ν-gap uncertainty. As a result, the choice of structure does not matter provided that the identification methods deliver optimal uncertainty sets rather than an uncertainty bound around a prefixed nominal model. Frequency-dependent closed-loop performance functions based on the uncertainty sets are again bounded by circles in the frequency domain, allowing for analytical expressions for worst-case performance and for the evaluation of the consequences of uncertainty for robust design. The results can be used to tune optimal experimental conditions in view of robust control design and in the further development of experiment-based robust control design methods.     

On the necessity of identifying the true parameter in adaptive LQ control

In adaptive control problems one may drop the requirement of identifying the true system in order to simplify the problem of control. It will be shown that in the adaptive LQ control problem this does not at all lead to an easier problem.