Worst-case System Identification Research Articles

Probabilistic Optimal Estimation With Uniformly Distributed Noise

The classical approach to system identification is based on stochastic assumptions about the measurement error, and provides estimates that have random nature. Worst-case identification, on the other hand, only assumes the knowledge of deterministic error bounds, and establishes guaranteed estimates, thus being in principle better suited for the use in control design. However, a main limitation of such deterministic bounds lies in their potential conservatism, thus leading to estimates of restricted use. In this paper, we propose a rapprochement between the stochastic and worst-case system identification viewpoints, which is based on the probabilistic setting of information-based complexity. The main idea in this line of research is to “discard” sets of measure at most ϵ, where ϵ is a probabilistic accuracy, from the set of deterministic estimates. Therefore, we are decreasing the so-called worst-case radius of information at the expense of a given probabilistic “risk.” In the case of uniformly distributed noise, we derive new computational results, and in particular we compute a trade-off curve, called violation function, which shows how the radius of information decreases as a function of the accuracy. To this end, we construct randomized and deterministic algorithms which provide approximations of this function. We report extensive simulations showing numerical comparisons between the stochastic, worst-case and probabilistic approaches, thus demonstrating the efficacy of the methods proposed in this paper.

Input design in worst-case system identification with quantized measurements

This paper addresses the problem of set membership system identification with quantized measurements. Following the work developed for binary measurements, the problem of optimal input design with multiple sensor thresholds is tackled. For a FIR model of order n, the problem is decomposed into n static gain problems. The one-step optimal input problem is solved both for equispaced and generic sensor threshold distribution. Moreover, the N-step optimal input problem for the case of equispaced thresholds is addressed, and a solution is provided under a suitable assumption on the sensor range and resolution. The obtained results allow us to construct an upper bound on the time complexity of the FIR identification problem for the case of equispaced thresholds. Numerical application examples are reported to show the effectiveness of the proposed algorithms.

Input design for worst-case system identification with uniformly quantized measurements

Optimal input design for system identification with quantized sensors is tackled in a worst-case setting. Under the assumption of uniformly quantized measurements, the paper addresses the one-step input design problem for static gains. Results are provided for both noise-free and noisy data and are instrumental for designing suboptimal inputs for FIR systems of arbitrary order. The input design problem over a time horizon of length N is also investigated, in order to characterize the time complexity of the considered identification problem.

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ℓq(X′) is a cover for a subset M of ℓP(X) if, for some 0 < α ≤ 1, ∥u * h∥ ∞ ≥ α ∥u∥q ∥h∥p for all h ∈ M. Covers of ℓ1 = ℓ1(R) are important in worst case system identification in ℓ1 and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ℓp(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.

A set-based approach for white noise modeling

This paper provides a new framework for analyzing white noise disturbances in linear systems; rather than the usual stochastic approach, noise signals are described as elements in sets, and their effect is analyzed from a worst-case perspective. The paper studies how these sets must be chosen to have adequate properties for system response in the worst-case, statistics consistent with the stochastic point of view, and simple descriptions that allow for tractable worst-case analysis. The method is demonstrated by considering its implications in two problems: rejection of white noise signals in the presence of system uncertainty and worst-case system identification.

Optimal nonparametric identification from arbitrary corrupt finite time series

Formulates and solves a worst-case system identification problem for single-input, single-output, linear, shift-invariant, distributed parameter plants. The available a priori information in this problem consists of time-dependent upper and lower bounds on the plant impulse response and the additive output noise. The available a posteriori information consists of a corrupt finite output time series obtained in response to a known, nonzero, but otherwise arbitrary, input signal. The authors present a novel identification method for this problem. This method maps the available a priori and a posteriori information into an uncertain of the plant, which comprises a nominal plant model, a bounded additive output noise, and a bounded additive model uncertainty. The upper bound on the model uncertainty is explicit and expressed in terms of both the l/sub 1/ and H/sub /spl infin// system norms. The identification method and the nominal model possess certain well-defined optimality properties and are computationally simple, requiring only the solution of a single linear programming problem. >

Worst-case system identification in [formula omitted]: Error bounds and optimal models : R.G. H~voort, pp 93–96

Worst-case system identification in [formula omitted]: Error bounds and optimal models : R.G. H~voort, pp 93–96

Stochastic and worst case system identification are not necessarily incompatible

Stochastic and worst case approaches to system identification are different and are usually treated separately. In this communique we investigate the effect that a projection operator has on the worst case behavior of estimates derived by stochastic identification algorithms. We show that under certain assumptions the projections of the stochastic estimates are convergent in the worst case setting. We illustrate this result by applying it to least squares and maximum likelihood algorithms.

On the time complexity of worst-case system identification

In this paper we treat a general worst-case system identification problem. This problem is worst-case with respect to both noise and system modeling uncertainty. We consider this problem under various a priori information structures. We determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst-case error bound. Our results are algorithm independent. We show that this minimum duration is prohibitively long. Based on our results, we suggest that worst-case (with respect to noise) system identification requires unrealistic amounts of experimental data. >

Suboptimal algorithms for worst case identification in H/sup ∞/ and model validation

New algorithms based on convex programming are proposed for worst case system identification. The algorithms are optimal within a factor of two asymptotically. Further, model validation, or data consistency, is embedded in the identification process. Explicit worst case identification error bounds in the H/sup /spl infin// norm are also derived for both uniformly and nonuniformly spaced frequency response samples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust approximate modelling of stable linear systems†

Robust approximation and worst-case approximate modelling of stable shift-invariant systems from corrupted transfer function estimates are studied in the H ∞ sense. Connections between the problem formulations of the present work and certain problems of worst-case system identification, notably the Helmicki-Jacobson-Nett problem formulation for identification in H ∞, are established. Issues of model set selection are addressed using the n-width concept: a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then-width sense. A notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well-posedness property for systems in the disk algebra. The existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way. A framework is given in which approximate models can be obtained as stable perturbations of the true system: these can be combined ...

Worst-Case System Identification in H∞: Error Bounds and Optimal Models

An identification procedure is developed which yields an upper bound for the H∞-norm of the model error for a given nominal model, using measurement data and a priori information consisting of a time domain bound on the noise and information about the decay rate of the pulse response of the model error. The upper bound is calculated by solving a set of linear programming problems. Moreover a procedure is presented to derive a new nominal model with reduced upper bound on the H∞-norm of the model error. This is performed in two steps, in the first step frequency response data are generated and in the second step a nominal model is identified with a curve-fit procedure, minimizing a maximum absolute value criterion function.