| 1. |
- Guo, Lei, et al.
(författare)
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Exponential Stability of General Tracking Algorithms
- 1994
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Tracking and adaptation algorithms are, from a formal point of view, nonlinear systems which depend on stochastic variables in a fairly complicated way. The analysis of such algorithms is thus quite complicated. A first step is to establish the exponential stability of these systems. This is of interest in its own right and a prerequisite for the practical use of the algorithm. It is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time-varying true ones. In this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms. This includes least mean squares, recursive least squares, and Kalman filter based adaptation algorithms. We show how stability of an averaged (linear and deterministic) equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes. We also give explicit conditions for exponential stability of the most common algorithms. The tracking performance of the algorithms is studied in a companion paper.
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| 2. |
- Guo, Lei, et al.
(författare)
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Performance Analysis of General Tracking Algorithms
- 1994
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Ingår i: Proceedings of the 33rd IEEE Conference on Decision and Control. - Linköping : Linköping University. - 0780319680 ; , s. 2851-2855 vol.3
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A general family of tracking algorithms for linear regression models is studied. It includes the familiar LMS (gradient approach), RLS (recursive least squares) and KF (Kalman filter) based estimators. The exact expressions for the quality of the obtained estimates are complicated. Approximate, and easy-to-use, expressions for the covariance matrix of the parameter tracking error are developed. These are applicable over whole time interval, including the transient and the approximation error can be explicitly calculated.
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| 3. |
- Guo, Lei, et al.
(författare)
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Performance Analysis of the Forgetting Factor RLS Algorithms
- 1992
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- An analysis is given of the performance of the standard forgetting factor recursive least squares (RLS) algorithm when used for tracking time-varying linear regression models. Three basic results are obtained: (1) the ‘P-matrix’ in the algorithm remains bounded if and only if the (time-varying) covariance matrix of the regressors is uniformly non-singular; (2) if so, the parameter tracking error covariance matrix is of the order O(μ + γ2/μ), where μ = 1 - λ, λ is the forgetting factor and γ is a quantity reflecting the speed of the parameter variations; (3) this covariance matrix can be arbitrarily well approximated (for small enough μ) by an expression that is easy to compute.
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| 4. |
- Guo, Lei, et al.
(författare)
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Performance Analysis of the Forgetting Factor RLS Algorithms
- 1993
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Ingår i: International journal of adaptive control and signal processing (Print). - : Wiley. - 0890-6327 .- 1099-1115. ; 7:6, s. 525-237
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Tidskriftsartikel (refereegranskat)abstract
- An analysis is given of the performance of the standard forgetting factor recursive least squares (RLS) algorithm when used for tracking time-varying linear regression models. Three basic results are obtained: (1) the ‘P-matrix’ in the algorithm remains bounded if and only if the (time-varying) covariance matrix of the regressors is uniformly non-singular; (2) if so, the parameter tracking error covariance matrix is of the order O(μ + γ2/μ), where μ = 1 - λ, λ is the forgetting factor and γ is a quantity reflecting the speed of the parameter variations; (3) this covariance matrix can be arbitrarily well approximated (for small enough μ) by an expression that is easy to compute.
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| 5. |
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| 6. |
- Guo, Lei, et al.
(författare)
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Tracking Performance Analysis of the Forgetting Factor RLS Algorithm
- 1992
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Ingår i: Proceedings of the 31st IEEE Conference on Decision and Control. - 0780308727 ; , s. 688-693 vol.1
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Konferensbidrag (refereegranskat)abstract
- The authors present a theoretical analysis for the performance of the standard forgetting factor recursive least squares (RLS) algorithm used in the tracking of time-varying linear regression models. Under some explicit excitation conditions on the regressors, it is shown that the parameter tracking error is on the order O(μ+γ2/μ), where μ=1-λ, λ is the forgetting factor, and γ is the quantity reflecting the speed of parameter variation. Furthermore, for a large class of weakly dependent regressors, simple approximations for the covariance matrix of this error are derived. These approximations are not asymptotic in nature: they hold over all time intervals and for all μ in a certain region.
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| 7. |
- Ljung, Lennart, 1946-, et al.
(författare)
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The Role of Model Validation for Assessing the Size of the Unmodeled Dynamics
- 1994
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Ingår i: Proceedings of the 33rd IEEE Conference on Decision and Control. - 0780319680 ; , s. 3894-3899 vol.4
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Konferensbidrag (refereegranskat)abstract
- There are two sources of errors in any identified model: 1) the bias error, due to too simple a model structure where all aspects of the true system cannot be described by any model within the used structure; and 2) the variance error, due to errors and disturbances in the measured data from which the model is constructed. The total model error is the sum of these two contributions, and the objective is to find a structure that makes this error small. While the variance error can be assessed by quite standard statistical methods, the bias error is far more difficult to evaluate. The present paper contains two results that relate to the size of the bias error to that of the variance error: 1) for a typical model that minimizes the total error, the bias error is dominated by the variance error; and 2) for a model that has passed a typical validation test, the bias error is again dominated by the variance error.
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