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 Objective: Derivation of H_{¥} controller
 Methods: Intuition and handwaving
 Background: State Feedback and Observer
 Problem Formulation and Solutions
 Bounded Real Lemma: another characterization
 An intuitive Proof

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 Consider a general LFT system
 Assumptions:
 (i) (A, B_{1}) is stabilizable and (C_{1},A) is
detectable;
 (ii) (A,B_{2}) is stabilizable and (C_{2},A) is
detectable;
 (iii) D_{12}*[C_{1}, D_{12}]=[0 I ]
 (iv)

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 (i) Together with (ii) guarantees that the two H_{2} AREs have
nonnegative stabilizing solutions.
 (ii) Necessary and sufficient for G to be internally stabilizable.
 (iii) The penalty on z = C_{1}x + D_{12}u includes a
nonsingular, normalized penalty on the control u. In the conventional H_{2}
setting this means that there is no cross weighting between the state
and control and that the control weight matrix is the identity.
 (iv) w includes both plant disturbance and sensor noise, these are
orthogonal, and the sensor noise weighting is normalized and
nonsingular.
 These assumptions can be relaxed.

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 Solution: $ K such
that T_{zw}_{¥}_{ }< g if and only if
 (i) X_{¥}_{
}³0
 X_{¥}_{
}A+A*X_{¥}_{ }+X_{¥}_{ }(B_{1}B_{1}*/g^{2}B_{2}B_{2}*)X_{¥}+C_{1}*C_{1}=0
 (ii) Y_{¥}_{
}³0
 AY_{¥}_{
}+Y_{¥}_{ }A*+Y_{¥}_{ }(C_{1}*C_{1}/g^{2}C_{2}*C_{2})Y_{¥}+B_{1 }B_{1}*=0
 (iii) r(X_{¥}_{ }Y_{¥}_{ })<
g^{2}
 Furthermore,
 where A_{¥}_{ }:=A+B_{1}B_{1}*X_{¥}/g^{2}+B_{2}F_{¥}+ Z_{¥}_{ }L_{¥}_{ }C_{2}
 F_{¥}_{
}:=B_{2}*X_{¥}_{ }, L_{¥}_{ }:=Y_{¥}_{ }C_{2}* , Z_{¥}_{ }=(IY_{¥}X_{¥}_{ }/g^{2})^{1}

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 Let z= G(s)w, G(s)=C(sIA)^{1}B Î RH_{¥}
 $ X =X*³0 such that X_{ }A+A*X+XBB*X
/g^{2}+C*C=0
and A+BB*X /g^{2}
is stable.
 Û
 Y=Y*³0 such that
YA*+AY+YC*CY /g^{2}+BB*=0 and A+YC*C /g^{2} is stable.

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 Note that the system equation can be written as
 State Feedback: u=Fx
 By Bounded Real Lemma: T_{zw}_{¥}_{ }< g Û $ X =X*³0 such that
 X(A+B_{2}F)+ (A+B_{2}F)*X+XB_{1}B_{1}*X
/g^{2}+(C_{1}+D_{12}F)*(C_{1}+D_{12}F)=0
 and A+B_{2}F+B_{1}B_{1}*X /g^{2} is stable.

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 (By completing square with respect to F to get) Û
 $ X =X*³0 such that
 XA+A*X+XB_{1}B_{1}*X /g^{2}X B_{2}B_{2}*X+C_{1}*C_{1}
 +(F+B_{2}*X )*(F+B_{2}*X)=0
 and A+B_{2}F+B_{1}B_{1}*X /g^{2} is stable.
 (Intuition suggests that F=B_{2}*X ) Û
 $ X =X*³0 such that
 XA+A*X+XB_{1}B_{1}*X
/g^{2}X
B_{2}B_{2}*X+C_{1}*C_{1}=0
 and AB_{2 }B_{2}*X +B_{1}B_{1}*X /g^{2} is stable.
 Þ F=F_{¥} and X =X_{¥}.
 Output Feedback: Converting to State Estimation
 Suppose $ a K
such that T_{zw}_{¥}_{ }< g
 Then x(¥ )=0 by stability (note also x(0) = 0)

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 Summary:
 If state is availabe; u=F_{¥} x
 worst disturbance: w_{*}= g^{}^{2}B_{1}*X_{¥} x

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