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Robust Control Theory and Applications Part 17

Tham khảo tài liệu 'robust control theory and applications part 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 628 Robust Control Theory and Applications where 7 - and TmflY - denote the smallest and largest singular values respectively. Suppose that the TF matrix is acoustically symmetric so that 7Yp ii cư and p 21 a2 7Yp42 a2 . We now have H W HpM 2 7Yp4i cu 2 1 cos 2 rA ÌA cos 2ttA-1A 1 32 where A denotes the interaural path difference given by A A12- Singular values can be found from the following characteristic equation 1 fc 2 cos2 2ttA ÌA 0. 33 By the definition of robustness the equalization system will be the most robust when cos 2ttA-1A 0 Hp cu is minimized and the least robust when cos 2ttA-1A 1 Hp cu is maximized Ward Elko 1999 . A similar analysis can be applied to acoustic energy density control. The composite transfer function between the two loudspeakers and the two microphones in the pressure and velocity fields becomes Herf cu p ll w Wp 21 w pc ÍHơ 11 cu pc ÍHưi cu p 12 w Wp 22 a2 pc H 12M pc ÍHơ 22M 34 where Hj cu is the frequency-domain matrix corresponding to Note that the pressure and velocity at a point in space X x j z p x and p x are related via jcvp V x Vp x 35 where V represents a gradient. Using this relation the velocity component for the X direction can be written as 36 where d and Ax denote the distance and the X component of the displacement vector between the With loudspeaker and the th control point respectively. Note that the velocity component for the y and z directions can be expressed similarly. Now we have where 2 Qcos 2 rA-1A Qcos 2ttA-1A 2 AxnAxi2 A1 11A1 12 AznAzi2 11 11 A Singular values can be obtained from the following characteristic equation 2 fc 2 Qcos ỉtĩả a 2 0. 37 38 39 H HriM 2 Q l From Eqs. 33 and 39 it can be noted that the maximum condition number of IHp cu equals to infinity while that of IHe i cf is 2 Q 2 Q when cosiUttA1 A 1. Eq. 38 also shows that the maximum condition number of the energy density field becomes smaller as A increases because Q approaches to 1. Now by comparing the maximum condition numbers Robust .