Dynamic locally-optimal control systems for objects with interval parameters and state delay

Consider the problem of dynamical locally-optimal control based on the observed output for discrete objects with interval parameters and delay in the state, described by the following difference equation: x(k + 1) = (A + Sigma(r)(i=1) A(i)theta(i))x(k) + ((A) over tilde + Sigma(r)(i=1) (A) over tilde (i)theta(i))x(k - h) + (B + Sigma(r)(i=1) B-i theta(i))u(k) + q(k); x(tau) = phi(tau), tau = -h, 1-h, 2-h, ... , 0; k = 0,1,2, ... , where x(k)is an element of R-n is a state vector, h > 0 is a positive integer time delay, u(k)is an element of R-m is a control input, A, A(i), (A) over tilde, (A) over tilde (i), B, B-i, i = (1, r) over bar are constant matrices of appropriate dimensions, q(k) is the Gaussian random sequence of input disturbances, theta(i) is uncertain parameters of interval type (-1 <= 0(i) <= 1). The measurement channel is represented by equation y(k) = Sx(k) + v(k), where S is the matrix of measurement channel, v(k) is the Gaussian random sequence of measurement errors. To solve the problem, we propose an algorithm, which is based on the optimization of the local criteria I(k) = M{(w(k + 1) - z(k))C-T(w(k + 1) - z(k)) + u(T) (k)Du(k)}, where w(k) = Hx(k) is the controlled output of the system, C = C-T >= 0 D = D-T >= 0 are weighting matrices, z(k) is an element of R-n is the tracking vector, described by equation z(k + 1) = Fz(k) + q(z) (k), where q(z)(k) is the Gaussian random sequence, F is a matrix. The control law of object is determined by the function of measured variables with time memory of the tracked signal and the dynamic element w(k): u(k) = K-0(k)w(k) + K-1(k)y(k) + K-2(k)y(k - h) + K-3(k)z(k). The formulas for calculating the optimal transfer coefficients K*(0)(k), K*(1)(k), K*(2)(k), K*(3)(k) are given. In this paper, the proposed synthesis algorithms of output control do not use an extension method of the state space. The asymptotic properties of the closed-loop system are obtained. For the square criterion J = lim(k ->infinity)M{parallel to x(k)-z parallel to(2)}, which defines the asymptotic accuracy of tracking, it is shown that J <= [(G + R)(2) + (g + r(2))(2) + tr (Q) over tilde][alpha(2)(1) + Phi(2)) + (alpha(2)(2) + Phi(2)(2))/1-(alpha(2)(1) + Phi(2)) + +2(alpha(1)alpha(2) + Phi Phi(2)) (G + R)(2) + (g + r(2))(2) + tr (Q) over tilde (1)/1-(alpha(2)(1) + Phi(2)) + +2(g + r(2)) (alpha(1) + alpha(2))(g + r(1)) + (Phi + Phi(2))(G + R)/1-alpha(1) + (g + r(1))(2) + (G + R)(2) + tr (Q) over tilde. So, we see that under natural restrictions on the class of dynamic systems, the method of locally optimal tracking on indirect measurements with errors provides asymptotic tracking with accuracy determined by the intensity of additive disturbances and errors in the observations, dynamic characteristics of a closed-loop system, values of the parameters of the object, and the transmission coefficients of the tracking control system. The comparison of the simulation results of the two control systems is given with: the optimal transfer coefficients, calculated with using the interval parameters; the transfer coefficients, calculated with using the nominal values of the parameters. The criterion of estimation of the quality of convergence of the state vector x(k) to the desired value z(k) shows that the average error of the deviation of the state vector x(k) of the tracking vector z(k) in the robust control is less than the control constructed on the nominal values of the parameters.