Filtration and diagnostics in discrete stochastic systems with jump parameters and multiplicative perturbations

Let an model with multiplicative perturbations and jump parameters be described by the equation m x(k +1) = A_r(k)x(k) + B_l(l)u(k) + £ A_sMl)x(k) Q_sMl) (k) + q_Y(k) (k), x(0) = x,, j=1 where x(k) e Rⁿ is the state vector, y = y(k) is the Markov chain with r states (y₁, y₂,...,y_r); x₀ is a random vector (the variance N_0i =M{(x₀ - x₀)(x₀ - x₀)^T / у = Y_i}, and the expectation x₀ = M{x₀} are assumed to be known) i = 1, r; A_y(k), B_y(k), A_sy(k) are the known matrices; u(k) e R^m is the known vector; q_y(lc)(k), 0_sy(_k) are the Gaussian random vector sequences with the following characteristics: M^k)} = 0, M^k)} = 0, M{ q_{j(l y}(k) q^ j 1© = 1 (k), k j} = Q* A, M^ )(k) )(j)/ y© = = y(k),k j} = ®_Y(k₎8_Ij. The probabilities of states of a jump process p_j(k) = P{y(1) = j}, j = 1,r satisfy the equations pj(k +1) = £p_t(k)p^j, pj(0) = pj,0, j = 1,r, i=1 where p_i j is the probability of the transition from state i to state j. The observation vector is determined by the formula y^(k ) = ^Sy(1 )^x(k) + ^VY(1 )^(k X where Vy^I ) is the Gaussian random sequence (M{vY_№)(k)} = 0, M{vY_№)(k) v^ _}(j) / y© = y(k), k j} = У^Ь^ ). The solution to the synthesis problem for filtering the state vector x(k) and identifying the value of the jump parameter Y(k) included in the description of a linear discrete system with additive and multiplicative perturbations is obtained. The problem is solved by introducing into the model of the system an unknown input vector that appears when the identification error of a jump parameter and the use of Kalman filtering with an unknown input.

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