Adaptive methods for piecewise linear filtering

  • Giovanni B. Di Masi
  • Marina Angelini
Part of the Progress in Systems and Control Theory book series (PSCT, volume 9)


A nonlinear discrete-time stochastic dynamical system is considered with piecewise linear drift coefficients and whose initial condition and disturbances are distributed according to finite mixtures of normal distributions. In particular the normal components of the mixtures relative to the state process have variances which vanish with a parameter ε.

For such system the nonlinear filtering problem is studied. It is shown that a suitable linear adaptive filtering problem can be constructed whose solution coincide, for vanishing e, with that of the original nonlinear problem.

The use of measure transformation techniques allows the derivation of the results under milder condition than those assumed so far in a similar context.


Piecewise Linear Stochastic Control Polynomial Growth Limit Problem Lebesgue Dominate Convergence Theorem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    V.Benes and I. Karatzas, Filtering for Piecewise Linear Drift and Observation, Proc. 20th Conf. on Dec. and Control (1981), 583–589.Google Scholar
  2. [2]
    G.B. Di Masi and W.J. Runggaldier, On measure transformations for combined filtering and parameter estimation in discrete time, Sys. and Control Letters 2 (1982), 57–62.CrossRefGoogle Scholar
  3. [3]
    G.B. Di Masi and W.J.Runggaldier, Asymptotic Analysis for Piecewise Linear Filtering, in Analysis and Optimization of Systems (A. Bensoussan and J.L. Lions eds.), Springer Verlag, Control and Info Sci. 111, 1988, 753–759.Google Scholar
  4. [4]
    G.B. Di Masi, W.J. Runggaldier, Piecewise linear stochastic control with partial observations,Proc. Imperial College Workshop on Applied Stochastic Analysis (M.H.A. Davis and R.J. Elliott eds.), Gordon and Breach, Stochastic Monograph Series, New York (to appear).Google Scholar
  5. [5]
    W.H.Fleming, D. Ji and E. Pardoux,(1988) Piecewise Linear Filtering with Small Observation Noise, in Analysis and Optimization of Systems (A. Bensoussan and J.L. Lions eds.), Springer Verlag, Control and Info Sci. 111, 1988, 752–759.Google Scholar
  6. [6]
    A.E.Kolessa, Recursive Filtering Algorithms for Systems with Piecewise Linear Nonlinearities,Avtom. Telemekh.. 5 (1986),48–55 (English Translation: 480–486).Google Scholar
  7. [7]
    R.S. Liptser and A.N. Shiryayev, Statistics of random processes, Springer-Verlag, New York, 1978.Google Scholar
  8. [8]
    A.Lasota and M.C. Mackey, Probabilistic properties of deterministic systems, Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
  9. [9]
    M. Loève, Probability theory, Van Nostrand Reihold Company, New York, 1963.Google Scholar
  10. [10]
    E. Pardoux and C. Savona, Piecewise Linear Filtering, in Stochastic Differential Systems, Stochastic Control Theory and Applications (W.H. Fleming and P.L. Lions eds.), Springer Verlag, IMA Volume in Mathematics and Applications 10, 1987.Google Scholar
  11. [11]
    C. Savona, C. Approximate Nonlinear Filtering for Piecewise Linear Systems, Systems and Control Letters 11 (1988), 327–332.CrossRefGoogle Scholar
  12. [12]
    F.L. Sims, D.G. Lainiotis and D.T. Magill, Recursive algorithm for the calculation of the adaptive Kalman filter weighting coefficients (Correspondence), IEEE Trans.Autom.Control AC-14 (1969), 215–218.Google Scholar
  13. [13]
    W.M. Wonham, Some applications of stochastic differential equations to optimal nonlinear filtering, SIAM J. on Control 2 (1965), 347–369.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Giovanni B. Di Masi
    • 1
    • 2
  • Marina Angelini
    • 3
  1. 1.Dipartimento di Matematica Pura ed ApplicateUniversità di PadovaPadovaItaly
  2. 2.CNR — LADSEBItaly
  3. 3.Venezia - MestreItaly

Personalised recommendations