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 background
 H_{¥} :
1984 workshop approach
 assumptions
 output feedback control
 a matrix fact
 inequality characterization
 connection between ARE and ARI (LMI)
 proof for necessity
 proof for sufficiency
 comments
 optimality and dependence on g
 H_{¥}_{ } controller structure
 example
 an optimal controller
 H_{¥}_{
} control: general case
 relaxing assumptions
 H_{2} and H_{¥}_{ } integral control
 H_{¥}_{
} filtering

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 Initial theory was SISO (Zames, Helton, Tannenbaum)
 NevanlinnaPick interpolation
 Operatortheoretic methods (Sarason, Adamjan et al, Ball
 Helton)
 Initial work handled restricted problems (1block and 2block)
 Solution to 2x2block problem
 (1984 HoneywellONR Workshop)

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 Solution approach:
 Parametrize all stabilizing
controllers via [Youla et al]
 Obtain realizations of the
closedloop transfer matrix
 Transform to 2x2 block general
distance problem
 Reduce to the Nehari problem and
solve via Glover
 Properties of the solution:
 Statespace using standard
operations
 Computationally intensive (many
Ric. eqns.)
 Potentially highorder
controllers
 Find solution < g , iterate for optimal

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 Consider a general LFT system
 Assumptions:
 (i) (A, B_{1}) is controllable and (C_{1},A) is
observable;
 (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 simplify the theorem statements and proofs, and can
be relaxed..

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 Solution: $ K such
that T_{zw}_{¥}_{ }< g if and only if
 (i) X_{¥}_{
}>0
 X_{¥}_{
}A+A*X_{¥}_{ }+X_{¥}_{ }(B_{1}B_{1}*/g^{2}B_{2}B_{2}*)X_{¥}+C_{1}*C_{1}=0
 (ii) Y_{¥}_{
}>0
 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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 [Packard, 1994] Suppose X, YÎR^{nxn}, and X=X*>0, Y=Y*>0. Let r
be a positive integer. Then there exist matrices
X_{12}ÎR^{nxr} , X_{2} Î R^{rxr} such
that X_{2}=X_{2}*, and
 if and only if

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 Proof.(Ü ) By assumption, there is a matrix X_{12}ÎR^{nxr} such
that XY^{1}=X_{12}X_{12}*. Defining X_{2}:=I_{r}
completes the construction.
(Þ )
Using Schur complements,
 Y=X^{1}+X^{1}X_{12}(X_{2}X_{12}*X^{1}
X_{12})^{1 }X_{12}*^{ }X^{1}
 ^{ }Inverting, using the matrix inversion lemma, gives
 Y^{1}=XX_{12}*X^{1}_{ }X_{12}*
 Hence, XY^{1}=X_{12 }X_{2}^{1}X_{12}*³0, and indeed,
 rank(XY^{1})=rank(X_{12
}X_{2}^{1}X_{12})£ r.

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 Lemma ARE: [Ran abd Vreugdenhil,
1988] Suppose (A,B) is controllable and there is an X = X* such that
 Q(X):=XA+A*X+XBB*X+Q<0.
 Then there exists a solution X_{+}>X to the Riccati equation
 X_{+}A+A*X_{+}+X_{+}BB*X_{+}+Q=0 (0.7)
 such that A+BB*X_{+} is antistable
 Proof. Let X be such that Q(X) < 0.
Choose F_{0} such that A_{0 }:= A BF_{0}
is antistable. Let X_{0}=X_{0}* solve
 X_{0}A_{0}+A_{0}*X_{0}F_{0}*F_{0}+Q=0.

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 Define G_{0}:=F_{0}+B*X. Then
 (X_{0}X)A_{0}+A_{0}*(X_{0}X)=G_{0}*G_{0}Q(X)>0
 and X_{0} > X (by antistability of A_{0}).
 Define a nonincreasing sequence of hermitian matrices {X_{i}}:
 X_{0} ³
X_{1} ³ … ³ X_{n1 }>X,
 A_{i}= A BF_{i}
, is antistable, i = 0,…., n1;
 F_{i}=B*X_{i1}, i=1, …, n1;
 X_{i}A_{i}+A_{i}*X_{i}=F_{i}*F_{i}Q,
i=0,1,…,n1. (0.8)
 By Induction: We show this sequence can indeed be defined.
 Introduce F_{n}=B*X_{n1} , A_{n}= A BF_{n}.
 We show that A_{n} is antistable. Using (0.8), with i = n1, we
get
 X_{n1}A_{n}+A_{n}*X_{n1}+Q F_{n}*F_{n}(F_{n}F_{n1})*(F_{n}F_{n1})=0.

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 Let G_{n}:=F_{n}+B*X.
Then
 (X_{n1}X)A_{n}+A_{n}*(X_{n1}X)=G_{n}*G_{n}Q(X
)+(F_{n}F_{n1})*(F_{n}F_{n1})>0
 which implies that A_{n }is antistable by Lyapunov stability
theorem since_{ } X_{n1}
X>0.
 Let X_{n} be the unique solution of
 X_{n} A_{n} +A_{n}*X_{n} = F_{n}*F_{n}
–Q. (0.9)
 Then X_{n} is hermitian. Next, we have
 (X_{n}X)A_{n}+A_{n}*(X_{n}X)=G_{n}*G_{n}Q(X
)>0
 (X_{n1}X_{n})A_{n}+A_{n}*(X_{n1}X_{n})=(F_{n}F_{n1})*(F_{n}F_{n1})³0
 Since A_{n }is antistable, we have X_{n1}³ X_{n }>X.
 Therefore, we have a nonincreasing sequence {X_{i}}.

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 Consider again the twomass/spring/damper system shown in Figure 0.1.
Assume that F_{1} is the control force, F_{2} is the
disturbance force, and the measurements of x_{1} and x_{2}
are corrupted by measurement noise:
 Our objective is to design a control law so that the effect of the
disturbance force F_{2} on the positions of the two masses x_{1}
and x_{2}, are reduced in a frequency range 0£w£2.

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 There exists a causal filter F(s)ÎRH_{¥} such that
J< g^{2}
if and only if J_{¥} Îdom(Ric) and
Y_{¥} =Ric(J_{¥} ) ³0
 where Y_{¥}
is the stabilizing solution to
