Locally-optimal control of discrete delayed control systems with incomplete information about state and perturbations

Model of object with delayed control is described by equation x(k+1) = Ax(k)+Bu(k-h)+Fs(k), x(0) = x(0), u(j) = psi(j), j =-h, -h+1,...,-1, where x(k) is an element of R-n is the state vector, u(k-h) is an element of R-m is the control vector, h is the time delay, s(k) is an element of R-n1 is the perturbation vector, x(0) and psi(j) (j =-h, -h+1,..., -1) are initial vector and initial function, A, B, and F are constant matrices. It is assumed that the observable vector w(x)(k) is an element of R-l, and w(x)(k) = H(x)x(k)+tau(x)(k), where H-x is the matrix of channel of observations, tau(x)(k) is the Gaussian random sequence. The perturbation model contains unknown parameters and is determined by the equation s(k+1) = (R(k)+Delta R(k))s(k)+f(k)+Delta f(k)+q(k), s(0) = s(0), where R(k) is the known matrix, f(k) is the known vector, Delta R(k) and Delta f(k) are some unknown matrix and vector, s(0) is the random vector of initial conditions independent of q(k), tau(k) and tau(x)(k); q(k) tau(k), tau(x)(k) are independent Gaussian random sequences with the known characteristics. Indirect observations of the vector perturbations are described by the model omega(k) = Phi s(k)+tau(k), where omega(k) is an element of R-m1 is the vector of observations, Phi is m(1) x n-matrix, tau(k) are random errors of observations. To solve the problem, we use the approach which is based on optimization of local criteria I(k) = M {(w(k+1) - z(k))C-T(w(k+1) - z(k)) + u(T)(k-h)Du(k-h)}, where w(k) = Hx(k) is the controlled output of the system, C = C-T >= 0 and D = D-T >= 0 are weight matrices, z(k) is an element of R-n is the tracking vector. The control is realized on the base of the Kalman filtering and extrapolation with considering the unknown input.