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Minimax Performance Limits for Multiple-Model Estimation

Olle Kjellqvist

Abstract

This article concerns the performance limits of strictly causal state estimation for linear systems with fixed, but uncertain, parameters belonging to a finite set. In particular, we provide upper and lower bounds on the smallest achievable gain from disturbances to the point-wise estimation error. The bounds rely on forward and backward Riccati recursions -- one forward recursion for each feasible model and one backward recursion for each pair of feasible models. We give simple examples where the lower and upper bounds are tight.

Minimax Performance Limits for Multiple-Model Estimation

Abstract

This article concerns the performance limits of strictly causal state estimation for linear systems with fixed, but uncertain, parameters belonging to a finite set. In particular, we provide upper and lower bounds on the smallest achievable gain from disturbances to the point-wise estimation error. The bounds rely on forward and backward Riccati recursions -- one forward recursion for each feasible model and one backward recursion for each pair of feasible models. We give simple examples where the lower and upper bounds are tight.
Paper Structure (20 sections, 7 theorems, 34 equations, 2 figures, 1 table)

This paper contains 20 sections, 7 theorems, 34 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Background
  4. Outline
  5. Notation
  6. Minimax performance limits
  7. Problem statement
  8. Forward Recursions
  9. Backward Recursions
  10. Examples
  11. Conclusions
  12. Future Work
  13. ACKNOWLEDGEMENTS
  14. Proofs
  15. Proof strategy
  16. ...and 5 more sections

Key Result

Proposition 1

$\gamma_N \geq \gamma_N^\star$ only if $P_i \preceq \gamma_N^2 I$ for all $i = 1, \ldots, M$.

Figures (2)

  • Figure 1: An illustration of the multiple-model estimation problem.
  • Figure 2: Numerically evaluated optimal performance levels, upper and lower bounds for the four system pairs considered in Section \ref{['sec:examples']}. Only stable and or distinguishable systems have bounded performance levels. In two pairs $\gamma_N^\star$ achieves the lower bound, and in Fig \ref{['fig:upperistight']} it approaches the upper bound.

Theorems & Definitions (13)

  • Remark 1
  • Proposition 1
  • Proposition 2: Minimax multiple-model estimator
  • proof
  • Theorem 1: Sufficient Condition
  • Theorem 2: Necessary Condition
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • ...and 3 more