In this chapter we discuss results related to approximate identification in H∞. The criteria for modelling and identification are formulated in terms of L∞or H∞norms. Emphasis is made on the construction of a model set by specifying bases in function spaces L2, H2 or in the disc algebra A(D). These bases include - besides the most widely used trigonometric basis - the recently introduced rational orthogonal bases and some wavelet bases. The construction of identification algorithms considered as bounded operators mapping measured noisy frequency response data to an element in the model space is discussed. Bounds on the operator norm effecting the approximation and the noise errors are given, too. It has been known that control design for dynamic systems usually requires the knowledge of an appropriate model of the system. These models can, in many cases, be derived from first principles but in more realistic situations from data that are the measured input/output signals of a system. Model construction from measured data is usually called system identification. The goal of system identification is to construct models from noise corrupted measured data such that the model and the system generating the data should be small under suitably chosen criteria. The choice of identification criteria and the parameterization of models should reflect the ultimate goal the model is intended to be used for and depends also on the available information about the experimental conditions, noise assumptions (stochastic, deterministic norm bounded), etc. The identification criteria can be formulated either in the time domain, for instance the prediction error criterion as in Chapter 4, or in the frequency domain with H2 and H∞criteria. Traditionally, the main approach to system identification has been based on stochastic assumptions explaining the errors between the actual system and its models. Properly parameterizing the models, the elaborated results provide a point estimate for the parameters of a nominal model and additional statistical properties characterizing the estimated parameters and the goodness of fit. The control design was based on the nominal model (applying the certainty equivalence principle) disregarding most of the statistical information provided by the identification. The appearance of the robust control paradigm in the past decade accompanied by the formal H∞analysis and design theory incorporated the modelling uncertainties into control design. This started as a completely deterministic approach with the design based on a family of models given by a nominal model and an uncertainty model, describing e.g. the modelling error (or the bound on the magnitude of the error) in the frequency domain of interest. The design has been usually formulated as an H∞optimization (e.g. minimization of certain operator norms) over a set of stabilization controllers. It was soon realized that existing methods of system identification are not able to provide initial data for robust control, and this inspired intensive research on both fields. The first concepts for a solution were published in the early 1990s by [123, 124]. This non-stochastic approach, usually referred to as worst-case identification for robust control, proposed to identify a nominal LTI model from frequency response data. The problems related to this research can be characterized as follows. The first subject considered is the modelling of uncertain systems and approximation of uncertain systems by a low complexity nominal model. The nominal model set is usually chosen as a finite dimensional subspace spanned by specific basis of one of the spaces H2, L2 or the disc algebra A(D); see their definitions below. The second problem is the identification of the uncertain system by determining a nominal model and bounds on the error that characterizes the uncertainty. The discrepancies between the system to be modelled and the nominal model is explained usually by two error sources. One of them is the approximation error and the second is the noise error. The approximation error is generally defined by the choice of approximation operator, whereas the noise error term is influenced both by the operator (more precisely the operator norm) and the norm of the noise that corrupts the measurements. A third group of problems, called validation/invalidation of models, comes from the question: If one specifies a nominal model and its uncertainty model, how it can be decided if these are consistent with measured information on the system? There are various approaches to the above problems both in the time and frequency domains. Recent overviews using information based complexity and the set-membership approach to modelling and identification can be found in [195-197] and . Concerning the choice of identification criteria, for worst-case identification in l1 see e.g. [50, 101, 114, 303]. Worst-case identification under H∞criterion appeared in a large number of papers including [123,124], [109,110], [246,247], , . Time-domain approaches to this problem appeared in  and . Closed-loop issues of identification for robust control were initiated by  using an LQG/LTR approach and were further discussed by [20, 305]. In [145, 146] a generic scheme was introduced for the joint identification/control design by showing that the identification and control errors are identical in this scheme. This allows to elaborate very powerful iterative tools to obtain high closed-loop performance. Model invalidation was discussed e.g. by [276, 277]. The subject of this chapter is the most closely related to worst-case identification under the H∞criterion by showing some new concepts and results that appeared recently in approximate modelling and identification. Our approach to build up the necessary tools on this area is based on the following concepts. The system to be modelled or identified from the data is supposed to be stable. The models should approximate this system uniformly under the H∞or L∞criterion if bounded noise assumption is applied. The problem is to find operators mapping the data set to a nominal LTI model such that the operator norm should be bounded to ensure convergence of the H∞or L ∞norm of the approximation error to zero by increasing model order. The noise error is required to be bounded, too. We put attention of choosing the model parameterization, i.e. by specifying the subspaces spanned by various bases that can be used to ensure the above requirements. The structure of the chapter is the following. The next section discusses the problem of approximate worst-case modelling and identification under the H ∞criterion and provides the basic definitions. It is followed by a discussion about approximate modelling and identification using the most widely used trigonometric basis in C2?, the space of 2π periodic functions continuous on the unit circle. This leads to model sets represented by weighted Fourier partial sum operators. These algorithms are called γ- summations of trigonometric Fourier series. Specific choices of the window function γ lead to well-known examples like the Fejér or de la Vallé e-Poussin summations. The model parameters are computed from frequency response data by using FFT. If the data are corrupted by L∞-norm bounded noise, the model set will be in a subspace of L∞. If a stable rational model with transfer function from the space A(D) is needed, one has to solve a Nehariapproximation problem. This typically two-step approach was discussed e.g. by [109,110] and , using Fejé r-summation in the first step. For a thorough treatment of this classical approach, one can consult . The properties and bounds of the operators generated by the general γ-summation are discussed in . Section 9.3 discusses the use of some recently introduced rational orthogonal bases in l2, H2 and L2. These bases were proposed by the authors of , [213, 214], and by . Special cases like the Laguerre and Kautz bases, discussed in [314, 315] and  will be also considered The new results are related to the discrete versions of the frequency-domain identification of these models. It will be shown that by introducing a special argument transform in the inner functions representing the generalization of the shift operator, these basis constructions can be relatively simply related to the trigonometric bases. This was shown in . The advantage coming from this property is that one can use the well-known FFT or DFT to compute the coefficients of the models. This enables the extension of the results obtained for the trigonometric bases to obtain approximate models and approximate identification algorithms under the H∞criterion, too. The next section discusses basis construction in the disc algebra (it is known that there is no basis in H ∞). The basis is derived from the Faber-Schauder or from the Franklin systems resulting in wavelet-like basis functions. This basis allows to use simple bounded linear operators to obtain approximate model sets. The parameters can be computed by biorthogonal functionals specified by the basis resulting in a very simple identification algorithm, see . Wavelet bases derived from frames and special rational bases proposed by  are considered briefly as well. This chapter is concluded by an application example of the frequency-domain identification of a MTI interferometer testbed.
ASJC Scopus subject areas