Local-optimal output control for discrete systems with state delays

Consider the problem of locally-optimal control based on the observed output for discrete objects with delay in the state described by the following difference equation: x(k +1) = Ax(k) + A x(k - h) + Bu(k) + q(k) , x(-t) = ф(т), T = -h, 1 -h, 2 -h,...,0; k = 0,1,2,... , where x(k)eR is a state vector, h > 0 is a positive integer time delay, u(k)eR is a control input, A,A ,B are the constant matrices of appropriate dimensions, q(k) is the Gaussian random sequence of input disturbances. The measurement channel is represented by equation y(k) = Sx(k) + v(k), where S is the matrix of the 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 following local criteria: I (k) = M{( w(k +1) - z(k )) C (w(k +1) - z(k)) + u (k) Du (k)}, where w(k) = Hx(k) is the controlled output of the system, C = C > 0 , and D = D > 0 are weighting matrices, z(k) e R is the tracking vector described by equation z(k +1) = Fz(k) + q (k), where q (k) is the Gaussian random sequence; F is a matrix. The control law of object is determined by the function of the measured variables with the time memory and tracked signal: u (k) = K (k)y(k) + K (k)y(k - h) + K (k)z(k). The formulas for calculating the optimal transfer coefficients K* (k), K* (k), K3 (k) are given. In this paper, the proposed algorithms synthesis of output control does not use the extension method of state space. The asymptotic properties of the closed-loop system are investigated. The estimators are obtained by the square criterion J = lim M{|| x(k) - z\\ }, k^да which defines the asymptotic tracking accuracy. It is shown that the estimator has the following form: J < а [-^r --L_.trQ] + --f r +-trQ] + 1 -а 1 -а 1 -а 1 -а 1 1 r + 2а1а [--- r + ---Щ ] + 4 + + trQ , 1 -а2 1 -а2 1 - а2 where r = \Bz\, r = BK *z , B (k) = BK*(k) - E , A =a , Ц =a , A(i) = A + BK*(i)S , Д(0 = A + BK*(i)S , us Q(k) = Q(k) + BK*(k)V(k)K* (k)B + BK *(k)V(k -h)K * (k)B , Q = K' B BK* V . The simulation results on the proposed methods are given.

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