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- Objective: Derivation of H¥ controller
- Methods: Intuition and hand-waving
- 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, B1) is stabilizable and (C1,A) is
detectable;
- (ii) (A,B2) is stabilizable and (C2,A) is
detectable;
- (iii) D12*[C1, D12]=[0 I ]
- (iv)
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- (i) Together with (ii) guarantees that the two H2 AREs have
nonnegative stabilizing solutions.
- (ii) Necessary and sufficient for G to be internally stabilizable.
- (iii) The penalty on z = C1x + D12u includes a
nonsingular, normalized penalty on the control u. In the conventional H2
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 ||Tzw||¥ < g if and only if
- (i) X¥
³0
- X¥
A+A*X¥ +X¥ (B1B1*/g2-B2B2*)X¥+C1*C1=0
- (ii) Y¥
³0
- AY¥
+Y¥ A*+Y¥ (C1*C1/g2-C2*C2)Y¥+B1 B1*=0
- (iii) r(X¥ Y¥ )<
g2
- Furthermore,
- where A¥ :=A+B1B1*X¥/g2+B2F¥+ Z¥ L¥ C2
- F¥
:=-B2*X¥ , L¥ :=-Y¥ C2* , Z¥ =(I-Y¥X¥ /g2)-1
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- Let z= G(s)w, G(s)=C(sI-A)-1B Î RH¥
- $ X =X*³0 such that X A+A*X+XBB*X
/g2+C*C=0
and A+BB*X /g2
is stable.
- Û
- Y=Y*³0 such that
YA*+AY+YC*CY /g2+BB*=0 and A+YC*C /g2 is stable.
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- Note that the system equation can be written as
- State Feedback: u=Fx
- By Bounded Real Lemma: ||Tzw||¥ < g Û $ X =X*³0 such that
- X(A+B2F)+ (A+B2F)*X+XB1B1*X
/g2+(C1+D12F)*(C1+D12F)=0
- and A+B2F+B1B1*X /g2 is stable.
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- (By completing square with respect to F to get) Û
- $ X =X*³0 such that
- XA+A*X+XB1B1*X /g2-X B2B2*X+C1*C1
- +(F+B2*X )*(F+B2*X)=0
- and A+B2F+B1B1*X /g2 is stable.
- (Intuition suggests that F=-B2*X ) Û
- $ X =X*³0 such that
- XA+A*X+XB1B1*X
/g2-X
B2B2*X+C1*C1=0
- and A-B2 B2*X +B1B1*X /g2 is stable.
- Þ F=F¥ and X =X¥.
- Output Feedback: Converting to State Estimation
- Suppose $ a K
such that ||Tzw||¥ < 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-2B1*X¥ x
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