P.M.J. Van den Hof

A generalized orthonormal basis for linear dynamical systems

In many areas of signal, system, and control theory, orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l/sub 2sup n/. To this end, use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, for example, the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e., to obtain a good approximation by retaining only a finite number of expansion coefficients. Consequences for identification of expansion coefficients are analyzed, and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients. >

Suboptimal Feedback Control by a Scheme of Iterative Identification and Control Design

In this paper a framework for an iterative procedure of identification and robust control design is introduced wherein the robust performance is monitored during the subsequent steps of the iterative scheme. By monitoring the performance via a model-based approach, the possibility to guarantee performance improvement in the iterative scheme is being employed. In order to monitor achieved performance (for a present controller) and to guarantee robust performance (for a future controller), an uncertainty set is used where the uncertainty structure is chosen in terms of model perturbations in the dual Youla parametrization. This uncertainty structure is shown to be particularly suitable for the control performance measure that is considered. The model uncertainty set can be identified by an uncertainty estimation procedure on the basis of closed-loop experimental data. To obtain performance robustness, robust control design tools are used to synthesise controllers on the basis of the identified uncertainty set.

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 generalized basis functions. The basis functions are orthogonal in H/sub 2/ and generalize the pulse, Laguerre and Kautz (1954) bases. The construction of the basis is considered and bias and variance expressions of the identification algorithm are discussed. The basis induces a new transformation (Hambo transform) of signals and systems, for which state space expressions are derived.<<ETX>>

Identification of probabilistic system uncertainty regions by explicit 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.

Cheapest open-loop identification for control

This paper presents a new method of identification experiment design for control. Our objective is to design the open-loop identification experiment with minimal excitation such that the controller designed with the identified model stabilizes and achieves a prescribed level of H/sub /spl infin// performance with the unknown true system G/sub 0/.

Multivariable closed-loop identification: from indirect identification to dual-Youla parametrization

Classical indirect methods of closed-loop identification can be applied on the basis of different closed-loop transfer functions. Here the multivariable situation is considered and conditions are formulated under which identified approximative plant models are guaranteed to be stabilized by the present controller. Additionally it is shown in which sense the classical indirect methods are generalized by a previously introduced identification method based on the dual-Youla parametrization. For stable controllers the two methods are shown to be basically equivalent to each other.Classical indirect methods of closed-loop identification can be applied on the basis of different closed-loop transfer functions. Here the multivariable situation is considered and conditions are formulated under which identified approximative plant models are guaranteed to be stabilized by the present controller. Additionally it is shown in which sense the classical indirect methods are generalized by a previously introduced identification method based on the dual-Youla parametrization. For stable controllers the two methods are shown to be basically equivalent to each other.

Multivariable least squares frequency domain identification using polynomial matrix fraction descriptions

In this paper an approach is presented to estimate a linear multivariable model on the basis of (noisy) frequency domain data via a curve fitting procedure. The multivariable model is parametrized in either a left or a right polynomial matrix fraction description and the parameters are computed by using a two-norm minimization of a multivariable output error. Additionally, input-output or element-wise based multivariable frequency weightings can be specified to tune the curve fitting error in a flexible way. The procedure is demonstrated on experimental data obtained from a 3 input 3 output wafer stepper system.In this paper an approach is presented to estimate a linear multivariable model on the basis of (noisy) frequency domain data via a curve fitting procedure. The multivariable model is parametrized in either a left or a right polynomial matrix fraction description and the parameters are computed by using a two-norm minimization of a multivariable output error. Additionally, input-output or element-wise based multivariable frequency weightings can be specified to tune the curve fitting error in a flexible way. The procedure is demonstrated on experimental data obtained from a 3 input 3 output wafer stepper system.

Multivariable feedback relevant system identification of a wafer stepper system

This paper discusses the approximation and feedback relevant parametric identification of a positioning mechanism present in a wafer stepper. The positioning mechanism in a wafer stepper is used in chip manufacturing processes for accurate positioning of the silicon wafer on which the chips are to be produced. The accurate positioning requires a robust and high-performance feedback controller that enables a fast throughput of silicon wafers. A control relevant set of multivariable finite dimensional linear time invariant discrete-time models is formulated and estimated on the basis of closed-loop experiments. The set of models is shown to be suitable for model-based robust control design of the positioning mechanism. This is illustrated by a successful design and implementation of a robust controller.

Identification of normalized coprime plant factors for iterative model and controller enhancement

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 the iterative identification/control design scheme as presented by Schrama (1992).<<ETX>>

Identification with Generalized Orthonormal Basis Functions - Statistical Analysis and Error Bounds

Abstract 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 H 2 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 few coefficients to be estimated. Explicit bounds are derived for the bias and variance errors that occur in the parameter estimates as well as in the resulting transfer function estimates.