Linear unbiased state estimation under randomly varying bounded sensor delay.

*(English)*Zbl 0944.93029The paper is about estimating the state of a plant from noise contaminated measurements which, in addition, are subject to a random delay. The authors show that in suitable dimensions the situation can be modelled as a discrete-time time-varying linear stochastic system with additive white noise (plant model) together with a read-out map,
\[
y(k)= C(k)[x(k), x(k-1)]^*+ D(k)[W(k), W(k- 1)]^*,
\]
which is perturbed by an independent white noise \(\{W(k)\}\) and for which, in addition, the matrices \(C(k)\) and \(D(k)\) are random (independent of the white noise processes) reflecting the random delay.

In this form the problem is about designing linear estimators for discrete-time linear system with additive and parametric noise. The authors derive the matrices \(K(k)\) and \(G(k)\) which determine the estimator \[ \widehat x(k+1)= K(k)\widehat x(k)+ G(k)y(k+ 1), \] to satisfy the requirements of unbiasedness and minimum variance for the estimation error \(x(k)-\widehat x(k)\).

In this form the problem is about designing linear estimators for discrete-time linear system with additive and parametric noise. The authors derive the matrices \(K(k)\) and \(G(k)\) which determine the estimator \[ \widehat x(k+1)= K(k)\widehat x(k)+ G(k)y(k+ 1), \] to satisfy the requirements of unbiasedness and minimum variance for the estimation error \(x(k)-\widehat x(k)\).

Reviewer: V.Wihstutz (Charlotte)

##### MSC:

93E10 | Estimation and detection in stochastic control theory |

93C55 | Discrete-time control/observation systems |

93B07 | Observability |

93E11 | Filtering in stochastic control theory |

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\textit{E. Yaz} and \textit{A. Ray}, Appl. Math. Lett. 11, No. 4, 27--32 (1998; Zbl 0944.93029)

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##### References:

[1] | Yaz, E.; Ray, A., Grammian assignment for stochastic parameter systems and their stabilization under randomly varying delays, Optimal control applications & methods, 16, 263-272, (1995) · Zbl 0840.93095 |

[2] | Ray, A., Output feedback control under randomly varying distributed delays, Journal of guidance, control & dynamics, 17, 4, 701, (1994) · Zbl 0925.93291 |

[3] | Nahi, N.E., Optimal recursive estimation with uncertain observation, IEEE trans. inform. theory, 15, 457-462, (1969) · Zbl 0174.51102 |

[4] | Rajasekaran, P.K.; Satyanarayana, M.; Srinath, M.D., Optimal linear estimation of stochastic signals in the presence of multiplicative noise, IEEE trans. aerospace systems, 7, 462, (1971) |

[5] | Tugnait, J.K., Stability of optimum linear estimators of stochastic signals in multiplicative noise, IEEE trans. aut. contr., 26, 757-761, (1981) · Zbl 0481.93062 |

[6] | DeKoning, W.L., Optimal estimation of linear discrete-time systems with stochastic parameters, Automatica, 20, 113-115, (1984) · Zbl 0542.93062 |

[7] | Yaz, E., Observer design for stochastic parameter systems, Int. jour. contr., 46, 1213-1217, (1987) · Zbl 0639.93063 |

[8] | Yaz, E., Implications of a result on observer design for stochastic parameter systems, Int. jour. contr., 47, 1355-1360, (1988) · Zbl 0652.93061 |

[9] | Yaz, E., Optimal state estimation with correlated multiplicative and additive noise and its application to measurement differencing, (), 317-318 |

[10] | Yaz, E., On the almost sure and Mean square exponential convergence of some stochastic observers, Ieee tac, 35, 935-936, (1990) · Zbl 0719.93011 |

[11] | Yaz, E., Robustness of stochastic parameter control and estimation schemes, IEEE trans. aut. contr., 35, 637-640, (1990) · Zbl 0709.93082 |

[12] | Yaz, E., Minimax state estimation for jump-parameter discrete-time systems with multiplicative noise of uncertain covariance, (), 1574-1578 |

[13] | Yaz, E., Estimation and control of stochastic bilinear systems with prescribed degree of stability, Ijss, 22, 835-843, (1991) · Zbl 0725.93078 |

[14] | Yaz, E., Full and reduced order observer design for discrete stochastic bilinear systems, IEEE trans. aut. contr., 37, 503-505, (1992) |

[15] | NaNacara, W.; Yaz, E., Recursive estimator for linear and nonlinear systems with uncertain observations, Signal processing, 62, 215-218, (1997) · Zbl 0908.93061 |

[16] | Ray, A., Performance analysis of medium access control protocols for distributed digital avionics, ASME journal of dynamic systems, measurement and control, 109, 370-377, (1987) |

[17] | Liou, L.-W.; Ray, A., A stochastic regulator for integrated communication and control systems: parts I and II, ASME journal of dynamic systems, measurement and control, 113, 604-619, (1991) |

[18] | Tsai, N.-C.; Ray, A., Stochastic optimal control under randomly varying distributed delays, International journal of control, 68, 1179-1202, (1997) · Zbl 0890.93082 |

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