Brett Ninness

A unifying construction of orthonormal bases for system identification

This paper develops a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of many previously studied orthonormal bases since the common FIR and recently popular Laguerre and two-parameter Kautz model structures are restrictive special cases of the construction presented here. However, in contrast to these special cases, the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject. Results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided.

Quantifying the error in estimated transfer functions with application to model order selection

Previous results on estimating errors or error bounds on identified transfer functions have relied upon prior assumptions about the noise and the unmodeled dynamics. This prior information took the form of parameterized bounding functions or parameterized probability density functions, in the time or frequency domain with known parameters. Here we show that the parameters that quantify this prior information can themselves be estimated from the data using a maximum likelihood technique. This significantly reduces the prior infor- mation required to estimate transfer function error bounds. We illustrate the usefulness of the method with a number of simula- tion examples. The paper concludes by showing how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and under-model- ing. Another simulation study compares our method to Akaikes well-known FPE and AIC criteria.

System identification of nonlinear state-space models

This paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state-space form. More specifically, a Maximum Likelihood (ML) framework is employed and an Expectation Maximisation (EM) algorithm is derived to compute these ML estimates. The Expectation (E) step involves solving a nonlinear state estimation problem, where the smoothed estimates of the states are required. This problem lends itself perfectly to the particle smoother, which provides arbitrarily good estimates. The maximisation (M) step is solved using standard techniques from numerical optimisation theory. Simulation examples demonstrate the efficacy of our proposed solution.

Estimation of model quality

This paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems. This is a very large field. We therefore concentrate on approaches that have been motivated by the need for reliable models for control system design. This will involve a discussion of efforts that go under the titles of ‘estimation in tH∞’, ‘worst-case estimation’, ‘estimation in l1’ and ‘stochastic embedding of undermodelling’. A central theme of this survey is to examine these new methods with reference to the classic bias/variance tradeoff in model structure selection.

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.

The fundamental role of general orthonormal bases in system identification

The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well-known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing certain new results on generalized Fourier series and the asymptotic properties of Toeplitz-like matrices in such a way that the new variance expressions presented here encompass pre-existing ones as special cases. Via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification. Namely, that orthonormal bases are much more than an implementational option offering improved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them.

Model Predictive Control Applied to Constraint Handling in Active Noise and Vibration Control

The difficulties imposed by actuator limitations in a range of active vibration and noise control problems are well recognized. This paper proposes and examines a new approach of employing model predictive control (MPC). MPC permits limitations on allowable control action to be explicitly included in the computation of an optimal control action. Such techniques have been widely and successfully applied in many other areas. However, due to the relatively high computational requirements of MPC, existing applications have been limited to systems with slow dynamics. This paper illustrates that MPC can be implemented on inexpensive hardware at high sampling rates using traditional online quadratic programming methods for nontrivial models and with significant control performance dividends.

Identification of Hammerstein-Wiener models

This paper develops and illustrates a new maximum-likelihood based method for the identification of Hammerstein-Wiener model structures. A central aspect is that a very general situation is considered wherein multivariable data, non-invertible Hammerstein and Wiener nonlinearities, and colored stochastic disturbances both before and after the Wiener nonlinearity are all catered for. The method developed here addresses the blind Wiener estimation problem as a special case.

Quantification of Uncertainty in Estimation

Models of physical processes rarely give an exact description of the systems response. Thus an important issue is the quantification of errors in model estimation due to model inadequacy. We show that this problem can be formulated using a Bayesian approach leading to simple formulae for model uncertainty. Techniques for minimizing the amount of computation are also discussed.

On Gradient-Based Search for Multivariable System Estimates

This paper addresses the design of gradient-based search algorithms for multivariable system estimation. In particular, the paper here considers so-called ldquofull parametrizationrdquo approaches, and establishes that the recently developed ldquodata-driven local coordinaterdquo methods can be seen as a special case within a broader class of techniques that are designed to deal with rank-deficient Jacobians. This informs the design of a new algorithm that, via a strategy of dynamic Jacobian rank determination, is illustrated to offer enhanced performance.