Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory 2nd Edition

1 Answers to exercises of chapters 6-8 The authors would like to express their appreciation to Jun Xu, Shuai Liu, Nan Xiao and Keyou You for their help in preparing the answers to the exercises in chapter 6-8. ANSWERS TO EXERCISES OF CHAPTER 6 1. Since T kkI,wehave 1 kIk F k FTT kk(1 kI F k FT)T k0(1) Applying Schur complements, the following inequality holds ! F k AT k DT k"!(1 k F k FT)1T k k1 kIk F k FTT k"! F k AT k DT k"0(2) Since kis the

4 6. The manipulation in the proof is similar to that in Theorem 6.4, so we only show the equivalence between the first constraint and (6.65). In order to simplify the notation, we use instead of k. By applying Lemma 6.2, the inequality is equivalent to # A+ D1 F$T# H+ D2 F$T # A+ D1 F$0 H+ D2 F0I = AT HT 0 I + 0 D1 D2 FT 0 0 T + FT 0 0 T 0 D1 D2 T 0 By applying Lemma 6.3, the above holds if AT HT A0 H0I + 0 D1 D2 1 0 D1 D2 T + FT 0 0 1 2 FT 0 0 T 0 which