A neural state estimator with bounded errors for nonlinear systems

Alessandri, A.; Baglietto, Marco; Parisini, Thomas; Zoppoli, R.

doi:10.1109/9.802911

Cited by 77 publications

(50 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The extended Ritz method with such bases was successfully tested on a variety of problems with admissible solutions dependent on a large number of variables: stochastic optimal control [64,65,66,80] and optimal estimation of state variables [3] in nonlinear dynamic systems with a large number of state variables, team optimal control [8], optimal control of freeway traffic [81], routing in large-scale communication networks [9,10], optimal fault diagnosis [5], etc. In these applications, admissible sets of variable-basis functions were used, for which the degree n necessary to guarantee a fixed approximation accuracy grows only polynomially with the number of variables of admissible solutions.…”

mentioning

confidence: 99%

“…For G formed by parameterized families of the form G = {g a : a ∈ A, g a ∈ G} with A ⊆ R p , this method was applied to a variety of optimization tasks considered in a series of papers [3,5,8,9,10,64,65,66,80] and in [81], where it was called the extended Ritz method. Here we use this term even more generally for an approximate optimization by the sequence of problems {M ∩ span n G}, where G is any subset of X.…”

mentioning

confidence: 99%

“…Also, a new branch of nonlinear approximation theory investigating approximation capabilities of neural networks was developed [11,12,21,28,38,45,50,51,52,53,54,55,56,57,58]. In a series of papers [3,5,8,9,10,64,65,66,80,81], a new method of approximate optimization was developed, called in [81] the extended Ritz method. In these papers, approximate solutions were used that were obtained over restrictions of sets of admissible solutions to linear combinations of all n-tuples of functions with varying "free" parameters, instead of linear combinations of first n functions from a basis with fixed ordering as in the classical Ritz method.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Error Estimates for Approximate Optimization by the Extended Ritz Method

Kůrková¹,

Sanguineti²

2005

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract. An alternative to the classical Ritz method for approximate optimization is investigated. In the extended Ritz method, sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n-tuples of functions from a given basis. This alternative scheme, called variable-basis approximation, includes functions computable by trigonometric polynomials with free frequencies, free-node splines, neural networks, and other nonlinear approximating families. Estimates of rates of approximate optimization by the extended Ritz method are derived. Upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable-basis functions, the modulus of continuity of the functional to be minimized, the modulus of Tikhonov well-posedness of the problem, and certain norms tailored to the type of basis. The results are applied to convex best approximation and to kernel methods in machine learning.Key words. functional optimization, rates of convergence of suboptimal solutions, (extended) Ritz method, curse of dimensionality, convex best approximation problems, learning from data by kernel methods AMS subject classifications. 49M15, 90B99, 90C90, 41A25, 68T05, 41A50

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Error Estimates for Approximate Optimization by the Extended Ritz Method

Kůrková¹,

Sanguineti²

2005

SIAM J. Optim.

View full text Add to dashboard Cite

show abstract

“…in [12], where a fully decentralized scheme is presented. In this paper we propose a distributed algorithm based on the concept of Moving Horizon Estimation (MHE), which has been proposed for discrete-time linear [13], [14], nonlinear [15], [16], [17], [18] and hybrid systems [19]. This approach has many advantages; first of all, the observer displays optimality properties, since a suitable minimization problem must be solved on-line at each time instant.…”

Section: Introductionmentioning

confidence: 99%

Distributed Moving Horizon Estimation for Linear Constrained Systems

Farina

Ferrari-Trecate

Scattolini

2010

IEEE Trans. Automat. Contr.

187

185

View full text Add to dashboard Cite

Abstract-This paper presents a novel distributed estimation algorithm based on the concept of moving horizon estimation. Under weak observability conditions we prove convergence of the state estimates computed by any sensors to the correct state even when constraints on noise and state variables are taken into account in the estimation process. Simulation examples are provided in order to show the main features of the proposed method.

show abstract

“…As regards the selection of the design parameters, it is obtained by means of a nonlinear-programming algorithm running offline [25], [26]. This is a major advantage over other neural approaches to estimation proposed in the literature (see, e.g., [27]- [31]), which rely on the online adaptation of the neural weights and so may involve a large number of computations in real-time applications.…”

Section: Introductionmentioning

confidence: 99%

Design of Asymptotic Estimators: An Approach Based on Neural Networks and Nonlinear Programming

Alessandri

Cervellera

Sanguineti

2007

IEEE Trans. Neural Netw.

View full text Add to dashboard Cite

Abstract-A methodology to design state estimators for a class of nonlinear continuous-time dynamic systems that is based on neural networks and nonlinear programming is proposed. The estimator has the structure of a Luenberger observer with a linear gain and a parameterized (in general, nonlinear) function, whose argument is an innovation term representing the difference between the current measurement and its prediction. The problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error. Toward this end, if a neural network is used to take on this function, the parameters (i.e., the neural weights) are chosen, together with the gain, by constraining the derivative of a quadratic Lyapunov function for the estimation error to be negative definite on a given compact set. It is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points. The gain is determined by solving a Lyapunov equation. The neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid-point constraints that are not satisfied. Techniques based on low-discrepancy sequences are applied to deal with a small number of sampling points, and, hence, to reduce the computational burden required to optimize the parameters. Numerical results are reported and comparisons with those obtained by the extended Kalman filter are made.

show abstract

A neural state estimator with bounded errors for nonlinear systems

Cited by 77 publications

References 30 publications

Error Estimates for Approximate Optimization by the Extended Ritz Method

Error Estimates for Approximate Optimization by the Extended Ritz Method

Distributed Moving Horizon Estimation for Linear Constrained Systems

Design of Asymptotic Estimators: An Approach Based on Neural Networks and Nonlinear Programming

Contact Info

Product

Resources

About