By Bruce A. Francis

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**Additional info for A cource in H Control Theory**

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Problem has already ily P of neighbouring C2] G" ] G 11 G 21 example [C1 (radius) function U w)11 < number (in terms plants in P. IR U w) I Stability under nominal Now define a fam- degree) matrices of poles in f1P satisfies the bound for all means internal proper proper real-rational of McMillan O:::;;w:::;;oo . proper K the robust stability For a real-rational in RHoo. plants to consist of all strictly the same 1. 2a from v l' v 2 to u, y all belong to RHoo. We saw in Chapter 1 that robust stability is guaranteed by a small gain con- dition.

Ii) is indeed nonsingular, (ii). e. its inverse exists as We have (3) Now is nonsingular because both M and V are. Also, since G 22 is strictly proper, we have that 1o 0 0] [ 1 K G 21 is nonsingular G 22 1 when evaluated at Thus both matrices on the right-hand The equations corresponding S =00: its determinant equals side of (3) are nonsingular. to Figure 2 are 1 at S =00. ('It. 1. Diagram for stability definition . 2. 4 30 Thus by Lemma 1 K stabilizes G iff -1 M - [~]u v -[OI]N But this is equivalent to (ii).

The function <1> is bounded if there exists a real number a such that II <1> x II ~ a II x II , x EX . The least such a is called the norm of <1>, denoted II <1> II. The following equations are not hard to derive: 11<1>11 = sup {11<1>x = sup {1I<1>x II : Ilx II ~ II : IIx II = I} I} . Such a bounded linear function is called an operator. Example 1. The Fourier transform, denoted Y, is an operator from Lz( -00,00) to Lz. 1 says that its norm equals 1. Example 2. 3 the direct sum Each function f in Lz(-oo,oo) f lELz(-oo,O] and f zELz[O,oo): has a unique decomposition f = f 1 +- f z with Ch.