By C. T. Leonides

Keep an eye on and Dynamic platforms quantity fifty five, half 1 contains assurance of the subsequent themes: numerical concerns in strong regulate layout options; loop move restoration for basic non-minimum part discrete time platforms; and set-induced norm dependent strong keep watch over ideas

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1 6 ) , let Vx : = the largest A-invariant subspace in k e r ( C i ) , (26) and let U be an orthogonal matrix U : = (Ui U2), where lmU2 = Vx. (27) Then C\V2 - 0 and AU2 = U2Ä with Ä stable since ( C i , A) is detectable. Following the development in Lemma 2, define the following transformed matrices: (28) Ύ ν Βλ where (Ci,A) = is observable. Finally, with these transformations define a re- duced version of ΗΊ: (30) Note that stabilizability of (Α,Βι) and (AyB2) imply that of (Α,Βι) and (4, # 2 ) , respectively.

Hence, /C is the set of all internally stabilizing feedback controllers for the system in Eq. ( 1 ) . Suppose K(s) G IC and define 7 := Halloo- (5) NUMERICAL ISSUES IN ROBUST CONTROL DESIGN TECHNIQUES 29 Then the objective in an Hoo control design is to find a controller K(s) € IC which minimizes 7 , the gain of the closed-loop system. T h e optimal gain is defined by lopt ^^ηί,ΙΙΓ,^ΙΙοο. A state-space-based solution to this problem is given in [5], [11],[18]. This approach relies on formulating the problem in terms of two Riccati equations which depend on a gain parameter, 7 .

N. 39 N U M E R I C A L ISSUES I N R O B U S T C O N T R O L D E S I G N T E C H N I Q U E S Proof: For any positive semi-definite matrices A and Ε the following identity holds [9] Xk(A) + XX(E) > Xk(A + E)> Xk(A) + Xn(E). Now choose A = ΧΊο and Ε = ΧΊ1 — ΧΊο and use the second inequality to get ^fc(Xy 0 + ^ 7 1 — - ^ 7 0 ) ^ ^ ( Χ γ 0 ) + λ η ( Χ 7 1 — X 1 Q) . T h e above implies that Χ^(ΧΊι) > λ & ( Χ γ 0 ) since Χη(ΧΊ1 - ΧΊ0) I > 0. So far we have shown that the individual Riccati solutions ΧΊ and ΥΊ are nonincreasing functions of 7.

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