Modulated MAP states optimal estimation under conditions of its partial observability | Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika – Tomsk State University Journal of Control and Computer Science. 2015. № 3(32).

Modulated MAP states optimal estimation under conditions of its partial observability

This paper considers the modulated MAP, which rate is a piecewise constant random process /(t) with two states: /(t) = / and /(t) = / (/ > / > 0). The time when the process /(t) remains at the i-th state, i - 1,2, depends on two random values: 1) the first random value has the exponential distribution function F/ - = 1 - e '', i = 1,2; when the i-th state ends, the process /(t) transits with the probability equal to one from the i-th state to the j-th state, i, j = 1,2 (i Ф j); 2) the second random value has the exponential distribution function F/ - = 1 - e , i = 1,2; when the i-th state ends, the process /(t) transits with the probability P ₁ (/ | /) from the i-th state to the j-th state (i Ф j) and a flow event occurs or /(t) transits with the probability P ₀ (/ | /) from the i-th state to the j-th state (i ^ j) but an event does not occur, or the process /(t) transits with the probability P ₁ (/ | /) from the i-th state to the i-th state and a flow event occurs. Here P ₁ (/j | /) + P ₀ (/j | /) + P ₁ (/ | /) = 1, i, j = 1,2, i Ф j. The block transition rate matrix for the process /(t) is as follows: -(а ₁ + /) а ₁ +/ ₁P ₀ (/ ₂| /) а ₂ +/ ₂P ₀(/ ₁ | / ₂) -(а ₂ +/ ₂) /1P1 (/J/1) /1P1 (/ ₂|/) / ₂ P1 (/1 | / ₂ ) / ₂ P1 (/ ₂|/ ₂) = 1D0 | Dj| . D = An event generates the period of time called the dead time, during which the flow cannot be observed. After this period ends, a new event also generates the dead time. Having only a sample (t ₁,., t _n) of events moments, we need to estimate the flow states. The optimal states estimation algorithm is as following: 1) at the initial moment t ₀ a priori probability л ₁ that /(t) is in / is calculated using the formula . =_а 2 +/2 [1 - P (/ ₂|/ 2)]_ _; а ₁ +а ₂ + / [1 -P ₁ (/ | /)] + / ₂[1 -P ₁ (/ ₂ | / ₂)] 2) in the interval (t ₀, t ₁) a posteriori probability w(/ ₁| t) is calculated using the formula | t) = 1 2 w(/1 0 + )-w2 1 w(/1 0 + ^--' ; 0 ) , w(/ | t0 + 0) = .1; ' w ₂ - w(/ ₁| t ₀ + 0)-[ - w(/ ₁| t ₀ + 0)]e A(w2 w1 Х"0 ' ^ l0 > 3) at event occurring moment t _k a posteriori probability w(/ ₁| t _k + 0) is calculated using the formula + / ₂ ^(/1 |/ ₂ ЫУК/ |/)-/ ₂ P1 (/1 |/ ₂ )]w(/1 | tk - 0) ' /2 [1 - P0 (М/)] + [[-/2-/1P0 (/J/) + / 2 P0 (М/ )]-w(/1 | tk - 0) where instead of w(/ ₁| t _k - 0) the value calculated on the formula w> ₂ -w(/ | tk + 0)]-w ₂[ -w(/ | tk + 0)e -w1 )-tk) W2 -w(/1 | tk + 0)-[ -w(/1 | tk + 0)e -w1 )(-tk) at t _k, k = 1,2,., is used; 4) in the interval (t _k, t _k + T _dead] the probability w(/ ₁| t) can be found as w(/ 11) = . + [w(/ | t _k + 0)-. ₁ ]• e k 5) in the interval (t _k + T _dead, t _k+1), k = 1,2,., the value w(/ ₁| t) can be calculated by formula used on the step 3, where instead of w(/ ₁| t _k + 0) the value w(/ ₁| t _k + T _dead) calculated on the step 4 is used. Then go to the step 3. The steps 3-5 are reiterated during observing time. Simultaneously with probability calculation we estimate the flow states: if w(/ ₁| t) > w(/ ₂| t) than / (t) = / ₁, otherwise / (t) = / ₂. The results show that the bigger а _i, i-1,2, the higher estimation error, and that the bigger quotient / ₁// ₂, the better estimation.

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Keywords

модулированный MAP-поток событий, оптимальное оценивание состояний, метод максимума апостериорной вероятности, непродлевающееся мертвое время, Modulated MAP, optimal state estimation, method of a posteriori probability maximum, non-extendable dead time

Authors

Name	Organization	E-mail
Berezin Dmitriy V.	Tomsk State University	berezin14@mail.ru
Nezhel'skaya Lyudmila A.	Tomsk State University	ludne@mail.tsu.ru

Всего: 2

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