Paul M.J. Van den Hof

Identification and control—closed-loop issues

An overview is given of some current research activities on the design of high-performance controllers for plants with uncertain dynamics, based on approximate identification and model-based control design. In dealing with the interplay between system identification and robust control design, some recently developed iterative schemes are reviewed and special attention is given to aspects of approximate identification under closed-loop experimental conditions.

System identification with generalized orthonormal basis functions

A least-squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function, where the expansion is in terms of recently introduced generalized basis functions. The basis functions are orthogonal in H2, and generalize the pulse, Laguerre and Kautz bases. One of their important properties is that, when chosen properly, they can substantially increase the speed of convergence of the series expansion. This leads to accurate approximate models with only a few coefficients to be estimated. Explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates.

An indirect method for transfer function estimation from closed loop data

Abstract An indirect method is introduced that is able to estimate consistently the transfer function of a linear plant on the basis of data obtained from closed loop experiments, even in the situation where the model of the noise disturbance on the data is not accurate. Moreover, the method allows approximate identification of the open loop plant with an explicit and tunable expression for the bias distribution of the resulting model.

Closed-loop issues in system identification

Abstract The identification of dynamical systems on the basis of data, measured under closed-loop experimental conditions, is a problem which is highly relevant in many (industrial) applications. Initiated by an emerging interest in the area called ‘identification for control’, classical prediction error identification methods have been extended to also handle the problem of identifying approximate models from closed-loop observations. In this paper the several procedures that have resulted from this research are reviewed and their characteristic properties are compared. Additionally it is discussed which role closed-loop identification can play in the identification of (optimal) models for (robust) control design.

Modelling and Identification with Rational Orthogonal Basis Functions

Abstract Decomposing dynamical systems in terms of orthogonal expansions enables the modelling/approximation of a system with a finite length expansion. By flexibly tuning the basis functions to underlying system characteristics, the rate of convergence of these expansions can be drastically increased, leading to highly accurate models (small bias) being represented by few parameters (small variance). Additionally algorithmic and numerical aspects are favourable. A recently developed general theory for basis construction will be presented, that is a generalization of the classical Laguerre theory. The basis functions are applied in problems of identification, approximation, realization, uncertainty modelling, and adaptive filtering, particularly exploiting the property that basis function models are linearly parametrized. Besides powerful algorithms, they also provide useful analysis tools for understanding the underlying identification/approximation algorithms.

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.

Identification of Normalised Coprime Plant Factors from Closed-loop Experimental Data

Recently introduced methods of iterative identification and control design are directed towards the design of high performing and robust control systems. These methods show the necessity of identifying approximate models from closed loop plant experiments. In this paper a method is proposed to approximately identify normalized coprime plant factors from closed loop data. The fact that normalized plant factors are estimated gives specific advantages both from an identification and from a robust control design point of view. It will be shown that the proposed method leads to identified models that are specifically accurate around the bandwidth of the closed loop system. The identification procedure fits very naturally into a recently developed the iterative identification/control design scheme based on H∞ robustness optimization.

Quantification of uncertainty in transfer function estimation: a mixed probabilistic—worst-case approach

Abstract In this paper an identification problem is solved which is directed towards the use of the identified model as a basis for robust control design. A procedure is presented to identify, on the basis of time domain measurement data, a reduced order finite impulse response (FIR) model together with an upper bound on the model error of the corresponding transfer function, using only minor prior information. We assume the measurement data to be contaminated with a stochastic noise disturbance with unknown spectral properties. By applying a procedure similar to Bartletts procedure of periodogram averaging, in conjunction with a periodic input signal, the statistics of the model error asymptotically can be obtained from the data. The model error consists of two parts: a probabilistic part, due to the stochastic noise disturbance, and a worst-case part, due to the unmodelled dynamics. The latter is explicitly bounded with a hard error bound, while for the former a confidence interval can be specified asymptotically. This enables an explicit trade-off between undermodelling (bias) and variance terms. The resulting error bound appears to be tight.

Hierarchical Long-Term and Short-Term Production Optimization

Model-based dynamic optimization of oil production has a significant potential to improve economic life-cycle performance, as has been shown in various studies. However, within these studies, short-term operational objectives are generally neglected. As a result, the optimized injection and production rates often result in a considerable decrease in short-term production performance. In reality, however, it is often these short-term objectives that dictate the course of the operational strategy. Incorporating short-term goals into the life-cycle optimization problem, therefore, is an essential step in model-based life-cycle optimization. We propose a hierarchical optimization structure with multiple objectives. Within this framework, the life-cycle performance in terms of net present value (NPV) serves as the primary objective and short-term operational performance is the secondary objective, such that optimality of the primary objective constrains the secondary optimization problem. This requires that optimality of the primary objective does not fix all degrees of freedom (DOF) of the decision variable space. Fortunately, the life-cycle optimization problem is generally ill-posed and contains many more decision variables than necessary. We present a method that identifies the redundant DOF in the life-cycle optimization problem, which can subsequently be used in the secondary optimization problem. In our study, we used a 3D reservoir in a fluvial depositional environment with a production life of 7 years. The primary objective is undiscounted NPV, while the secondary objective is aimed at maximizing short-term production. The optimal life-cycle waterflooding strategy that includes short-term performance is compared to the optimal strategy that disregards short-term performance. The experiment shows a very large increase in short-term production, boosting first-year production by a factor of 2, without significantly compromising optimality of the primary objective, showing a slight drop in NPV of only ?0.3%. Our method to determine the redundant DOF in the primary objective function relies on the computation of the Hessian matrix of the objective function with respect to the control variables. Although theoretically rigorous, this method is computationally infeasible for realistically sized problems. Therefore, we also developed a second, more pragmatic, method relying on an alternating sequence of optimizing the primary- and secondary-objective functions. Subsequently, we demonstrated that both methods lead to nearly identical results, which offers scope for application of hierarchical long-term and short-term production optimization to realistically sized flooding-optimization problems.

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 @n-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.