DGKF Problem StatementGiven generalized plant˙x(t)z(t)y( t)=A B1B2C10 D12C2D210x(t)w(t)u(t)with assumptions• the pair (A, B1) is stabilizable, the pair (A, C1) is detectable.• orthogonality conditions on dataDT12C1D12=0 I,B1D21DT21=0IConsider controllers of the form˙η(t)u(t)=AcBcCcDcη(t)y( t)that render the closed-loop system internally stable. Let Tzwdenote theclosed-loop mapping from w to z.Given γ, determine conditions for the ex istence of a controller so thatkTzwk∞< γ.For any γ f or which this is possi ble, parametrize all such controllers.183DGKF Solvability ConditionsGiven γ, define Ha milto nian matricesH∞:=A γ−2B1BT1− B2BT2−CT1C1−AT, J∞:=ATγ−2CT1C1− CT2C2−B1BT1−ATheorem 3: There exists a stabilizing controller such that kTzwk∞< γ ifand only if the following 3 conditions hold:1. H∞∈ dom(Ric), and X∞:= Ric (H∞) 0.2. J∞∈ dom(Ric), and Y∞:= Ric (J∞) 0.3. ρ (X∞Y∞) < γ2.Moreover, when these co nditions hold, one such controller, called the centralcontroller, isKcen:=ˆA∞−Z∞L∞F∞0whereˆA∞:= A + γ−2B1BT1X∞+ B2F∞+ Z∞L∞C2F∞:= −BT2X∞, L∞:= −Y∞CT2, Z∞:=I − γ−2Y∞X∞−1.Note that the first condition (for example) is that ther e ex ist a matrix X∞=X∞ 0 satisfying A +γ−2B1BT1− B2BT2X∞Hurwit z, andATX∞+ X∞A + X∞γ−2B1BT1− B2BT2X∞+ CT1C1= 0This is the exact condi tion in the State Feedback/Full Informat ion case de-rived earli er.184DGKF Lemma/Theorem DependenciesIn the table below, the dependencies of each lemm a and theorem on the oth-ers is given. This aids in preparing a strateg y for r eading DGKF in a sensiblemanner. The onl y difference is that we proved result FI 4 independently,using different techniques.Result Depends on...L1 Independent, in Riccati NotesL2 Independent, in Riccati NotesL3 Independent, in Riccati NotesL4 Independent, HomeworkL11 Independent, Balanced realizatio n homeworkL12 Independent, easy to verifyL14 Independent, easy to verifyL15 Independent, proof is straightforwa rdL16 Independent, easy to verifyP3 Independent, easyP4 Independent, easy, but subtleFI 4 Independent, difficultL5 L16, e asy PBHL6 L14, e asyL13 L4, L1 2 UnneededL17 L14, e asy verificationL10 L1, L6 , easyL18 L6, L1 4, L6, easyP1 L17, straightforwardP2 L3, L1 7, L13, UnneededFI 1 L17FI 2 L17, P1FI 3 prelimary calculation in T1FI 5 L18, L 15 (++)FC 1-5 FI, dualityDF 1-5 FI, P3, P4OE 1-5 DF, dualityL7 OE, PBHL8 FI 4L9 L5, L7 , OE (?), L10, L15, L18T1 L17, OE1, OE2T2 L17, OE3T3 L10, OE4, OE5, L9; L8, L9, L10, OE4T4 L9,
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