THE SKEWED S. S. V. і/

Д is now split as Д = diag(A,A2), where Ді and Д2 are two mixed structured perturbations. The skewed 1. 1.1. в( M) is defined as:

v{M) = 1 /тіп(к / ЗД = diag(Ax, к A2) with А j € BAj

and det(I — MA) = 0)

= 0 if no (k, Ді, Д2) exists (1-29)

When computing ц, the unit ball BA is expanded (or shrunk) by factor к until the matrix I – MA becomes singular for a structured perturbation inside кВА. When computing u, the unit ball BA2 (in the space of perturbations Д2) is expanded (or shrunk) by factor k, but the structured perturbation A remains now inside its unit ball ЯДі.

Proposition 4..1 illustrates that it is possible under mild conditions to

compute the exact value of v (resp. of /x) by computing recursively the exact value of д (resp. of д). When computing the exact value of u, it is moreover possible to use either a fixed point or a dichotomy search.

Proposition 4.. 1 Let mi the dimension of matrix Д*.

a/ if (Мп) < 1, then u(M) is the unique limit of the fixed point

iteration:

THE SKEWED S. S. V. і/

b/ if (Мц) < 1, then v(M) is the unique zero of the monotonous function:

THE SKEWED S. S. V. і/

c/ if Мд(М) > /хдх (Mu), then /хд(М) is the unique limit of the fixed point iteration:

THE SKEWED S. S. V. і/

Proof: see (Fan and Tits, 1992) for points a/ and c/ (with some technical arrangement because of the potential discontinuity of the mixed s. s.v.). Point b/ is deduced from point a/ using (a > 0):

/х(-Я) = -/x(tf) (1-33)

a a

Note however that u(aH) Ф av(H).

Remark: u(M) takes an infinite value if and only if /хд, (Мц) > 1 (Fan and Tits, 1992). It is indeed easily remarked that if /хд^Мп) > 1, then there exists a perturbation Д = diag(Ai,0) which renders the matrix 1-М A singular andsatisfies <т(Ді) < 1, so that v(M) = +oo. When ap­plying p, analysis to the standard interconnection structure M(s)-A(s), the nominal closed loop is assumed to be asymptotically stable, so that the s. s.v. vд(M(jut) ) can only take finite values. On the contrary, an infinite value can be obtained for the v measure.

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