Optimal Designs for Steady-State Kalman Filters

  • Guillaume SagnolEmail author
  • Radoslav Harman
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 122)


We consider a stationary discrete-time linear process that can be observed by a finite number of sensors. The experimental design for the observations consists of an allocation of available resources to these sensors. We formalize the problem of selecting a design that maximizes the information matrix of the steady-state of the Kalman filter, with respect to a standard optimality criterion, such as D- or A-optimality. This problem generalizes the optimal experimental design problem for a linear regression model with a finite design space and uncorrelated errors. Finally, we show that under natural assumptions, a steady-state optimal design can be computed by semidefinite programming.


Kalman Filter Linear Matrix Inequality Information Matrix Semidefinite Programming Optimal Design Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The research of the second author was supported by the VEGA 1/0163/13 grant of the Slovak Scientific Grant Agency.

Supplementary material

331057_1_En_17_MOESM1_ESM.pdf (130 kb)
(PDF 130 kB)


  1. 1.
    Amouroux M, Babary JP, Malandrakis C (1978) Optimal location of sensors for linear stochastic distributed parameter systems. In: Distributed parameter systems: modelling and identification. Springer, Berlin, pp 92–113 CrossRefGoogle Scholar
  2. 2.
    Ben-Tal A, Nemirovski A (1987) Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Society for industrial mathematics, vol 2 Google Scholar
  3. 3.
    Bhatia R (2008) Positive definite matrices. Princeton University Press, Princeton Google Scholar
  4. 4.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge CrossRefzbMATHGoogle Scholar
  5. 5.
    Curtain RF, Ichikawa A (1978) Optimal location of sensors for filtering for distributed systems. In: Distributed parameter systems: modelling and identification. Springer, Berlin, pp 236–255 CrossRefGoogle Scholar
  6. 6.
    Grant M, Boyd S (2010) CVX: Matlab software for disciplined convex programming, version 1.21.
  7. 7.
    Gupta V, Chung TH, Hassibi B, Murray RM (2006) On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage. Automatica 42(2):251–260 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Henrion D (2011) Semidefinite representation of convex hulls of rational varieties. Acta Appl Math 115(3):319–327 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kailath T, Sayed AH, Hassibi B (2000) Linear estimation. Prentice Hall, New York Google Scholar
  10. 10.
    Mourikis AI, Roumeliotis SI (2006) Optimal sensor scheduling for resource-constrained localization of mobile robot formations. IEEE Trans Robot 22(5):917–931 CrossRefGoogle Scholar
  11. 11.
    Pukelsheim F (1993) Optimal design of experiments. Wiley, New York zbMATHGoogle Scholar
  12. 12.
    Rafajłowicz E (1984) Optimization of measurements for state estimation in parabolic distributed systems. Kybernetika 20(5):413–422 MathSciNetzbMATHGoogle Scholar
  13. 13.
    Sagnol G (2012) Picos, a python interface to conic optimization solvers. Technical report 12-48, ZIB.
  14. 14.
    Sagnol G (2013) On the semidefinite representation of real functions applied to symmetric matrices. Linear Algebra Appl 439(10):2829–2843 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Simon D (2006) Optimal state estimation: Kalman, H , and nonlinear approaches. Wiley, New York CrossRefGoogle Scholar
  16. 16.
    Singhal H, Michailidis G (2010) Optimal experiment design in a filtering context with application to sampled network data. Ann Appl Stat 4(1):78–93 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Dept. OptimizationZuse Institut BerlinBerlinGermany
  2. 2.Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovakia

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