Optimal estimate of the states of a generalized synchronous flow of second-order events under conditions of incomplete observability

The paper deals with a generalized synchronous flow second-order events accompanying random process A(t), which is an unob-servable piecewise constant process with two states S₁ and S₂ . Hereinafter, it is understood the i th state of process A(t) as the state S,, i = 1,2 . The duration of interval between the flow events at the _i th state is determined by random variable ] = min(4,⁽¹⁾, 4⁽²⁾), where random variable 4⁽¹⁾ is distributed according to the law F_i⁽¹⁾(t) = 1 -e^-Ait, random variable 4⁽²⁾ is distributed according to the law F⁽²⁾(t) = 1 - г~^а,'; 4⁽¹⁾ and 4⁽²⁾ are independent of each other, i = 1,2 . At the moment when a flow event occurs, process A(t) transits from the , th state to the j th either with probability p⁽¹⁾(A_y. | A_;) or with probability р⁽²⁾(A_y. | A) depending on the value of random variable ], i, j = 1,2 , i Ф j . At the moment when a flow event occurs, process A(t) stays in the i th state either with probability p⁽¹⁾(A- | A) or with probability p⁽²⁾(A | A,) depending on the value of random variable i,, i = 1,2 . Wherein, p⁽¹⁾(A · | A) + P⁽¹⁾(A | A) = 1, P⁽²⁾(A | A) + P⁽²⁾(A | A) = 1, i, j = 1,2 , i Ф j . Thus, the duration of interval between the flow events in the i th state of process A(t) is a random variable with the exponential distribution function F(t) = 1 -^A +а'ⁱ>^t, i = 1,2 . In the sequel it is assumed that the state S₁ (the first state) of random process A (t) takes place, if A(t) = A₁, and the state S₂ (the second state) of random process A (t) takes place, if A(t) = A₂ (A₁ > A₂ > 0 ). We consider the situation, where each event registered at the moment t_k generates the period of time T , called a dead time, during which other flow events are lost, and upon its completion, an occurring event also causes the period of non-observability of the flow. It is necessary to estimate the state of random process X (t) at the moment t, having a sample of the moments of occurrence of events t₁, t₂,... in the observed flow. The algorithm of optimal estimation for the states of a generalized synchronous flow of second-order events with unextendable dead time is as follows: 1) at the initial instant t₀ = 0 a priori probability of the first state я of the process X(t) is calculated using explicit expression _{я =}_X₂p⁽¹⁾(X| X₂) + a₂ P1⁽²⁾(X11X₂)_. ¹ XP® (X₂1X1) + aP® (X₂1X1) + X₂P⁽¹⁾(X11X₂) + a₂P^<2)(A,11X₂)' 2) k = 0, at any moment t of the interval (t₀,t₁) a probability w(X₁ 11) is computed according to formula •(X1 +a1 -X2 -a₂)(t-tk) w(XJ t_k + 0)e w(X 11) =----л-;-тггг, w(X₁ 11₀ + 0) = я,; ^(1|) 1 - w(X | h + 0) + w(X | h + 0)e"^(X1+"1^-X2-"^2)(t-tk^ ⁽110 ) 1; at the moment t₁ calculations are made for determination w (X₁11₁) = w(X₁11₁ - 0) using the same formula; 3) k increases by 1; for k = 1a posteriori probability is recalculated according to formula w(X | t_k + 0) = w /[(X₂ + a) + w(X | t_k - 0)(X + a - x₂ - a₂)], W = X^P⁽¹⁾(X1 | X₂) + a₂P⁽²⁾(X1 | X₂) + w(X1 | tk-0XXP°4X | X1) + ap⁽²⁾(X1 | XJ-X₂PP⁽¹⁾(X1 | X₂)-aP⁽²4X | X₂)]; | ^ + 0) is the initial value for w (X₁ 11) at the next step; 4) k = 1, at any moment t of the half-interval (t₁, t₁ + T] a probability w (X₁ 11) is calculated according to w(X111) = я + [w(X11 t_k + 0) -rcje"^a('-'^k >, a = x P⁽¹⁾(X₂1X) + aP⁽²⁾(X IX) + X P⁽¹⁾(X IX) + a₂P⁽²⁾(X | X); here w(X₁11 = t₁ + T) is the initial value for w (X₁ 11) at the next step of the algorithm; 5) k = 1, at any moment t of the (t₁ + T,t₂) a posteriori probability w(X₁ 11) is calculated according to _w(X I h + T)e"^(t1+^{a1 -X2-a2)(t-tk-T)}_, ^w( 11 ~ 1 - w(X 14 +T) + w(X 14 + T)e~^(X1+^{a1 -X2-a2)(t-tk-T);} w(X₁ 11₂) = w(X₁ 11₂ - 0) can be found using the same formula when t = t₂; 6) the algorithm goes to step 3, after that steps 3-5 are repeated sequentially for k = 2 and so on. Simultaneously with the calculation process of probability w (X₁ 11) , the estimation of the flow state is made according to criterion of a posteriori probability maximum: if w(X₁11) > w(X₂11), then x(t) = X, else X(t) = X. The numerical results of described statistical experiments conducted on the simulation model demonstrate the possibility of sufficiently qualitative estimation of the states of a generalized synchronous flow of second-order events in conditions of a constant dead time.

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