By Hidenori Kimura

1 Introduction.- 2 components of Linear platforms Theory.- three Norms and Factorizations.- four Chain-Scattering Representations of the Plant.- five J-Lossless Conjugation and Interpolation.- 6 J-Lossless Factorizations.- 7 H-infinity regulate through (J, J')-Lossless Factorization.- eight State-Space strategies to H-infinity keep watch over Problems.- nine constitution of H-infinity keep watch over

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**Extra info for Chain-scattering approach to h[infinity] control**

**Sample text**

8. Jordan Forms 45 respectively. The student should verify that P-'AP~ [~ ~ ~l Example 4. Find a basis for R4 which reduces A= [°0-1° -2 -1]° 1 2 1 1 001 1 1 to its Jordan canonical form. J= 1 (1) - VI - [-~]° ° (1) V2 - [-~]° 1 span Ker(A - AI). We next solve (A - AI)V = Cl v~l) + C2V~1). These equations are equivalent to X3 = C2 and Xl can therefore choose Cl = 1, C2 = 0, Xl = 1, X2 = VI(2) + X2 + X3 + X4 = Cl. X3 We = ( 1,0,0,0 )T ° (with V~I) = (-1,1,0, O)T); and we can choose Cl Xl = -1, X2 = X4 = and find v 2(2) ° = X4 = and find 1 = ( -1,0,1,0 )T .

0 -b]° Case IV. B = b The phase portrait for the linear system (2) in this case is given in Figure 4. Cf. Problem 1(d) in Problem Set 1. The system (2) is said to have a center at the origin in this case. Whenever A has a pair of pure imaginary complex conjugate eigenvalues, ±ib, the phase portrait of the linear system (1) is linearly equivalent to one of the phase portraits shown in Figure 4. Note that the trajectories or solution curves in Figure 4 lie on circles Ix(t)1 = constant. In general, the trajectories of the system (1) will lie on ellipses and the solution x(t) of (1) will satisfy m ::; Jx(t)J ::; M for all t E R; cf.

0 ~j (2) 1 >. for>. one of the real eigenvalues of A or of the form Iz D D o ~j (3) with a D = [b for>. = a + ib -b]a' 12 = [10 0]1 and 0 = [~ ~] one of the complex eigenvalues of A. This theorem is proved in Coddington and Levinson [CjL] or in Hirsch and Smale [HjS]. The Jordan canonical form of a given n x n matrix A is unique except for the order of the elementary Jordan blocks in (1) and for the fact that the l's in the elementary blocks (2) or the 12 's in the elementary blocks (3) may appear either above or below the diagonal.