Peter S. C. Heuberger

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. >

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.

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 dynamic models in complex networks with prediction error methods : basic methods for consistent module estimates

Abstract 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.

Approximate system identification using system based orthonormal functions

The problem of approximate system identification is addressed. The use of prefilters that are based on a special class of system based orthonormal functions is proposed. It is shown that every linear finite dimensional time invariant discrete time system gives rise to two sets of orthonormal functions and that both can be used as a basis of the space l/sub 2/. The derivation of these functions, to be considered as a generalization of the Laguerre polynomials, is based on the properties of discrete time all-pass functions. Transformation of the input/output signals of a linear system in terms of these functions leads to new system descriptions, and new possibilities arise for the construction of approximate identification methods, with favorable properties allowing the use of simple estimation techniques and a systematic choice of prefilters.<<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.

Crucial aspects of zero-order hold LPV state-space system discretization

In the framework of Linear Parameter-Varying (LPV) systems, controllers are commonly designed in continuous-time, but implemented on digital hardware. Additionally, LPV system identification is formulated exclusively in discrete-time, needing structural information about the plant, which is often provided by first principle continuous-time models. These imply that LPV system discretization is an important issue for both system identification and controller implementation. Discretization approaches of LPV state-space systems are introduced and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling of the input-output signals and the scheduling parameter of the system. Criteria to choose appropriate sampling times with the investigated methods are also presented.

Orthonormal Basis Functions in Time and Frequency Domain: Hambo Transform Theory

The class of finite impulse response (FIR), Laguerre, and Kautz functions can be generalized to a family of rational orthonormal basis functions for the Hardy space H2 of stable linear dynamical systems. These basis functions are useful for constructing efficient parameterizations and coding of linear systems and signals, as required in, e.g., system identification, system approximation, and adaptive filtering. In this paper, the basis functions are derived from a transfer function perspective as well as in a state space setting. It is shown how this approach leads to alternative series expansions of systems and signals in time and frequency domain. The generalized basis functions induce signal and system transforms (Hambo transforms), which have proved to be useful analysis tools in various modelling problems. These transforms are analyzed in detail in this paper, and a large number of their properties are derived. Principally, it is shown how minimal state space realizations of the system transform can be obtained from minimal state space realizations of the original system and vice versa.

Identification of Dynamic Models in Complex Networks With Prediction Error Methods: Predictor Input Selection

This paper addresses the problem of obtaining an estimate of a particular module of interest that is embedded in a dynamic network with known interconnection structure. In this paper it is shown that there is considerable freedom as to which variables can be included as inputs to the predictor, while still obtaining consistent estimates of the particular module of interest. This freedom is encoded into sufficient conditions on the set of predictor inputs that allow for consistent identification of the module. The conditions can be used to design a sensor placement scheme, or to determine whether it is possible to obtain consistent estimates while refraining from measuring particular variables in the network. As identification methods the Direct and Two Stage Prediction-Error methods are considered. Algorithms are presented for checking the conditions using tools from graph theory.

Minimal partial realization from generalized orthonormal basis function expansions

A solution is presented for the problem of realizing a discrete-time LTI state-space model of minimal McMillan degree such that its first N expansion coefficients in terms of generalized orthonormal basis match a given sequence. The basis considered, also known as the Hambo basis, can be viewed as a generalization of the more familiar Laguerre and two-parameter Kautz constructions, allowing general dynamic information to be incorporated in the basis. For the solution of the problem use is made of the properties of the Hambo operator transform theory that underlies the basis function expansion. As corollary results compact expressions are found by which the Hambo transform and its inverse can be computed efficiently. The resulting realization algorithms can be applied in an approximative sense, for instance, for computing a low-order model from a large basis function expansion that is obtained in an identification experiment.