By Cornelius T. Leondes
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III OUTPUT COVARIANCE CONSTRAINT PROBLEM 52 GUOMINGG. ZHU AND ROBERTE. SKELTON Suppose that the continuous system is sampled at ~t different rates 1/T~, I/T2 . . . 1/T. t unique integers Pl, P2 ..... p, and a unique real number A such that T, = p,A, where the lowest common multiple of p~, p~ ..... p, is equal to one . 1) z(k,n) - M p x ( k , n ) + v ( k , n ) , for n = 0,1 ..... 1). 1), then the pair (k,n) represents the time axis at (pk + n)A. ) are zero mean cyclo-stationary processes with periodic covariance matrices W ( n ) and V(n), or are s disturbances defined below, oo p-I IIw(, )11---E E wr (k'n)W-~ (n)w(k,n) < oo, W(n) > 0 k=O n=O where w = [ w pr' vr]r is an s disturbance and W(n) ( n = 0 , 1 .
In this example much more robustness can be guaranteed for perturbations at the time n - - 2 , than for either n = 0, or n = 1. 1561 This section presents several robustness bounds related to input disturbances, and time-varying both structured and unstructured parameter variations for discrete periodically time-varying and multirate systems. 3a). By setting the period parameter p equal to one, one gets robustness bounds for the discrete time-invariant case. Therefore, these bounds are the generalization of the bounds for discrete time-invariant systems.
The line minimization utilizes golden section and quadratic interpolation. At every iteration, the matrices (Ok), (Gk) denote the direction and the gradient matrices at that iteration. The gradient of the cost function with respect to the gain matrices (Kj) is computed using finite differences employing step sizes relative to the individual magnitudes of the gain matrix. Although either forward or central differences can be utilized in computing the gradient matrix, central differences produce more reliable gradient computation but forward differences require half the number of computations.