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) = x0, u(j) = \\i(j), j = -h,-h +1,...,-1, where x(k) e R is the state vector, u(k -h) e R is the control vector, h is the time delay, s(k) e R is the perturbation vector, x₀ and \(j) (j = -h,-h + 1,...,-1) are initial vector and initial function, А, В, and F are constant matrices. It is assumed that the observable vector w_x (k) e R, and w_x (k) = H_xx(k) + x_x (k), where H_x is the matrix of channel of observations, x_x (k) is the Gaussian random sequence. The perturbation model contains unknown parameters and is determined by the equation s(k +1) = (R(k) + IR(k))s(k) + f (k) + if (k) + q(k), s(0) = s₀, where R(k) is the known matrix, f(k) is the known vector, IR(k) and if (k) are some unknown matrix and vector, s₀ is the random vector of initial conditions independent of q(k), x(k) and x_x (k); q(k), x(k), x_x (k) are independent Gaussian random sequences with the known characteristics. Indirect observations of the vector perturbations are described by the model 0 and D = D > 0 are weight matrices, z(k) e R is the tracking vector. The control is realized on the base of the Kalman filtering and extrapolation with considering the unknown input.

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