By I. A Ibragimov

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**Extra info for Independent and stationary sequences of random variables**

**Example text**

G. Meyer, " A New Class of Shift Varying Operators, Their Shift-invariant Equivalents and Multirate Digital Systems ", IEEE Transactions on Automatic Control, vol. AC-35, pp. 429-433, 1990 24. Y. J. Patton, " A Robust Approach to Multirate Controller Design Using Eigenstructure Assignment ", American Control Conference 1990, San Diego, California, pp. 945-951 25. N. Schiavoni, P. Colaneri and R. Scattolini, "' LQG Optimal Control of Multirate Sampled Data Systems ", IEEE Transactions on Automatic Control This Page Intentionally Left Blank Output Covariance Constraint Problem for Periodic and Multirate Systems Guoming G.

E[k]+ 2Y_E[k]MMu_M[k] + u__~t[k]RMu~[k] } kro--O (76) where (k) refers now to the model rate sampling instants and the multirate cost weighting matrices QM, MM, RM are given by Qar =diag { Q R , . ,RB }Si (79) Note that t~s definition of the multirate performance index resulted from the definition of the modified expanded state vector X-E and our design goal for a model rate controller. : with Z kTo=O + + (80) 20 IANNIS S. APOSTOLAKIS AT ^ QF. = A EQMA E ME = ~ rEMM + A^ EQMIE2 r BM (81) r r RE - BMIE2QMIE2BM + 2B~ITE2MM + RM (83) (82) As a direct result of our approach to include the cross weighting term MB to mapping a given continuous cost function to a multirate one and the multirate sampling structure introduced, the resulting discrete expanded control weighting matrix RE includes a non-symmetric term.

4704. 57) To'T, "T2 < 1, any AA(k,0), AA(k,1), AA(k,2) asymptotically stable. 31) is The flexibility of choosing 7, can be demonstrated as follows. Suppose we desire the most robustness at sample time n = 1. 3062 > 0; 0"504372 - 0. 4704 > 0. 0960, maximizing robustness at the inter-sample time n = 1. 2 shows the maximal guaranteed robustness at any one inter-sample time. In this example much more robustness can be guaranteed for perturbations at the time n - - 2 , than for either n = 0, or n = 1.