Identification of discrete time systems with random jump parameters and incomplete information.pdf Estimation and identification problems are relevant for different systems. As an example of such systems, one can consider, for example, economic systems [1, 2], energy systems [3, 4], flight systems [5], communication systems [6, 7]. Such problems occupy a special place in the problem of fault detection [8-10]. In [11], the problem of filtering and simultaneous diagnostics of a jump parameter for discrete systems with multiplicative perturbations was considered. In this paper, we consider the problem of simultaneous extrapolation and identification of a state with a jump parameter described by a Markov chain, which is included in the description of a linear stochastic system. The solution was obtained using the separation principle, Kalman extrapolator and the vector of estimates of the unknown input [12-15]. It is proposed to select a filter transmission matrix based on minimizing the sum of quadratic forms of estimation errors. The identification problem is solved in the conditions of incomplete information about the observation (there is an unknown input in the observation channel model). A numerical example of solving the problem of extrapolation and identification of a linear system with Markov jump parameter is given. 1. Problem statement Consider the following linear discrete-time stochastic system with a jump parameter: x(k +1) = Ai(k)x(k) + A(k)U(k) + qyik)(k), x(0) = x0, (1) where x(k) e Rn denotes the state of the system, u (k) e Rm denotes the known input, X0 is a random vector, Ay(k) and A(k) are matrices of corresponding dimensions; у = y(k) is a jumping parameter not available to observations (Markov chain with r states y1,...,yr); qy(k}(k) are random perturbations with characteristics: E{ qJ(k)(k)} = ° E{ qyk)(k) qlk) СО I y©=y(k) k j}=• Here E{ •} denotes the mathematical expectation, T denotes matrix transposition and 5kJ is Kronecker delta. The probability of states of the jump process Pj (k) = P{y(k) = j}, j = 1, r satisfies the equation r _ Pj(k+o=Z Pi(k) Pi, j, Pj(0)=Pj,o, j=1 r, (2) i=1 48 Identification of discrete time systems with random jump parameters and incomplete information where ptd is the probability of transition from the state i to the state j for one step, Pj,0 is the initial probability of the j-th state. An observation vector with incomplete information is: y(k) = SY(k)x(k) + HY(kMk) + VY(k) (kI (3) where y(k) is an unknown input, HY(k) is a matrix, vY(k) (k) is the Gaussian random sequence independent of Яу(к)(!с) xo and Y(k) with characteristics: E{vY(k)(k)} = 0, E{vY(k)(k)vTk)(j)| Y© = Y(k),k 0, satisfying the equations: r _ L, (k) = (A - K (k)S, )T (X P.jLj (k + 1))(A - K (k)S,) +1, L, (T) = H, i = 1, j=1 where I is an unit matrix, H > 0 is some matrix. Let us sum over k = t,T -1 the finite differences of the function W(k, N (k)), taking into account formula (12): T-1 T-1 T-1 XAW(k,N(k)) =X[W(k +1,N(k +1))- W(k,Ni(k))] =Xtr[Ni(k + 1)Li(k +1)k=t k=t k=t -N (k)L (k) - [Qi + K (k)YiKi (k)т + о, (k)]L (k)]. (13) On the other hand, this expression can be represented as follows: T -1 XAW(k, N (k)) =W(t+1, N (t+1)) - W(t, N (t))+... k=t +W (T, N (T)) - W (T -1, N (T -1)) = tr N (T)L (T) - T-1 (14) -trN(t)Lt (t) - trX[Q, + K(k)VK(k)T+Q(k)]Lt (k). k=t Add to the formula (10) the difference of the right-hand sides (13) and (14). Given that this difference is zero, then criterion (10) will take the form: J[0,T, i] = X tr Nt (k) -X tr N (k) L (k) + k=0 k=0 +Xtr[(At -K(k)St)(XPijNj(k))(A -K(k)StУ +Q + K(k)VK(k)T]Lt(k +1). _ (15) k=0 j=1 Applying the rules of differentiating the trace function from the matrix [15], we calculate the derivatives 8J[0,T, i] 6 £4 £4 - =-{X tr N. (k) - X tr N. (k)Lt (k) + 6K (k) 6K (k) X0 i ( * X i ( * i ( * k=0 T-1 +X tr[(A - K (k)St )(X р,, jNj (k))(A - K (k)St f + k=0 j=1 50 Identification of discrete time systems with random jump parameters and incomplete information +Q, + K (k)ViKi (kf]L, (k +1)} = X 2[-L, (k +1) Ai (X PijNj (k))Sj 4=0 J=1 r +Li (k + 1)K (k )Si (X Pi, jNj (k ))Sj + L (k +1) K (k )Vi ]. (16) j=1 Equating this derivative to zero and assuming that each summand over i is equal to zero, we obtain formula (9) for determining the matrix Ki(k). Now calculate the finite difference of the Lyapunov function AW (k, N (k)) = W (k +1, Nt (k +1)) - W (k, Nt (k)) = T = tr Nt (k +1) + tr X Q + K (t)VK (t)T+Q (t)]Li (t) t=k+1 -tr N (k) - trX [Qi + K (t)VK (t)T +Q (t)]L (t) = t=k = tr N (k +1) - tr N (k) - tr[Q + K (k )ViKi (k)T+Q (k)]Lt (k). (17) Since the Lyapunov function (11) is positive, and its finite difference (17), specifying the matrices Q i (t) > 0 accordingly, is negative, this guarantees the stability of the extrapolator (7). The theorem is proved. 4. Stationary extrapolator In this case, the optimized criterion has the form J[0,i] = lim1e|X eT(k)e(k) | y(0) = y,), T t {k=o J (18) the transfer matrices K are constants and are determined from the following matrices algebraic equations: N = (A - K,S, )(XPjN3 )(A - K,S, )T + Qt + K (k)V,K, (k)T J j=1 (19) K = A, (XPi,jNj)ST[sSi (XPuNj )SJ + V]-1. (20) j=1 j=1 So, the stationary extrapolator takes the form x(k +1) = Ax(k) + f(k) + Kt (y(k) - Six(k) - Ht(k)), x(0) = x0. (21) Note that if there are positive definite solutions Nt (i = 1, r) of the matrices equation (19), then from the condition Q + > 0 follows the validity of Theorem 1.6 [17], and this means the stability of the stationary extrapolator (21). 5. Unknown input and jump parameter estimation As an algorithm for estimating an unknown input f (k) and (k) we will use LSM-estimates; in this case, an estimate can be constructed on the basis of minimizing the additional criterions [12] under the assumption that the value of the jump parameter is known (у = yt): Ъ(№) = X{lly(t) -St^HW +1 V(t -1) ||W2}, (22) G2 (f (k)) = X {I y(t) - H

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