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# hw3 - J.M Gonalves c Winter 2004 CDS 213 Robust Control...

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J.M. Gon¸ calves Winter 2004 CDS 213 - Robust Control Homework # 3 Date Given: March 4, 2004 Date Due: Optional P1. Do question 7.3, DP, p. 232. P2. Do question 7.9, DP, p. 234. Show that one such controller is K ( s ) = A + B 1 B * 1 X - B 2 B * 2 X - ( I - Y X ) - 1 Y C * 2 C 2 ( I - Y X ) - 1 Y C * 2 - B * 2 X 0 . Now do question 7.6, DP, p.234. P3. Let M and N be suitably dimensioned matrices and let Δ be a structured uncertainty. Prove or disprove: 1. μ Δ ( M ) = 0 = M = 0. 2. μ Δ ( M 1 + M 2 ) μ Δ ( M 1 ) + μ Δ ( M 2 ). 3. μ Δ ( αM ) = | α | μ Δ ( M ). 4. μ Δ ( I ) = 1. 5. μ Δ ( MN ) σ ( M ) μ Δ ( N ). 6. μ Δ ( MN ) σ ( N ) μ Δ ( M ). P4. 1. Let Δ = Δ 1 0 0 Δ 2 with Δ i structured uncertainties. Show that μ Δ M 11 M 12 0 M 22 = max { μ Δ 1 ( M 11 ) , μ Δ 2 ( M 22 ) } . 2. Let M = 0 M 12 M 21 0 be a complex matrix and let Δ = Δ 1 0 0 Δ 2 . Show that μ Δ ( M ) = p σ ( M 12 ) σ ( M 21 ) . (Hint: Take Θ = θ 0 0 1 in Proposition 8.15, p. 256.) 3. Let M = M 11 M 12 M 21 M 22 be a complex matrix and let Δ = Δ 1 0 0 Δ 2 . Show that p σ ( M 12 ) σ ( M 21 ) - max { σ ( M 11 ) , σ ( M 22 ) } ≤ μ Δ ( M ) p σ ( M 12 ) σ ( M 21 )+max { σ ( M 11 ) , σ ( M 22 ) } . P5. Consider the feedback system shown in Figure 1. Consider the uncertain system P = P 0 (1 + W 1 Δ 1 ) + W 2 Δ 2 , || Δ i || < 1 , i = 1 , 2 where P 0 is an uncertain description and P and P 0 have the same number of poles in the RHP, and W 1 and W 2 are stable.

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