Robust extrapolation in discrete systems with random jump parameters and unknown input

We consider the model of an object that is described by a discrete equation x(k+1) = Ax(k) + U(k)+f(k)+q_y(k), x(0) = x₀, where x(k) e R" is the state vector, y(k) is a Markov chain with n statesy_l,y₂,...,y„; U_y(k) is known input;f(k) is the unknown input; X0 is a random vector (the variances are assumed to be known N_0I = M{(x₀ -x₀)(x₀ -x₀)^T / у = у}, i = Щ, x₀ = M{x₀} ); A_f is given matrix; q_Y(k) is random disturbance with the following characteristics M{q_y(k)} = 0 , M{q_y(k)qT(k)} = . The observation channel has the form y(k) = S._fx(k) + v._f(k), where v_y(k) is random errors of observations independent of the q_T (k) with M{v_y (k)} = 0, M{v_y (k) vT(k)} = . According to the information obtained at time k, it is required to find an estimate of the state of the forecast x(k +1) by minimizing the following criterion: ^Tf n n J[0;Tf ] = M {[££ Pi (k )e^T(k )R (k )e(k) + £ p, (T_f )e\T_f )Ц (T_f )e(T_f)] /у(0) = y₀}, k=0 i=1 i =1 where e(k) = x(k) - x(k), R (k) > 0 and L_t (T_f) > 0 are weight matrices, у₀ is the initial value of the variable y. The structure of the extrapolator is defined by the equation x(k +1) = Ax(k) + U(k) + f (k) + K(k)(y(k) - S_yx(k)), x(0) = x, where K(k) is the matrix of transmission coefficients to be determined. Theorem. Assume that there exist positive definite matrices N_t and L_t being solutions of the two-point boundary problem: n N_t(k +1) = (At -K(k)StPt;Nj(k))(At -K(k)St)^T + Qt + K(k)VtK(k)^T, N(0) = N₀. j=1 n L(k) = (A - K(kS)^T(XPjLj (k+1))(A - K(kS)+R, L_t(Tf)=Lt,. j=1 Then the vector ct(K(k)), composed with using of rows of the matrix K(k), is determined by the formula: ct(K (k)) = (k + 1)[L (k +1) ® St (k ))St^T + Ц (k +1) ® V ])^-1 x '=1 j=1 x ct(]Tpt(k + 1)Lt(k +1)AtC£pt,jNj(k))St^T). t=1 j=1 Thus, the synthesis algorithm robust extrapolator is developed that allows to define estimate of the state vector for discrete linear system with random jump parameter described by a Markov chain with a finite number of states. The filter transmission coefficients are proposed to be chosen by minimizing the sum of the quadratic forms of extrapolation errors, while averaging over the probabilities of the state of the jump parameter using recursive estimation algorithms with an unknown input. The extrapolation problem is considered when the value у is identified both accurately and with errors.

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