Elements of Feedback Control

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Similarity transformation
same system as()
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Example
diagonalized
decoupled
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Invariance
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Controllability
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Example
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Controller Canonical Form
Completely Controllable
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Controllability
Only need to check this for eigenvalues
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Controllability
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PBH test for diagonal case
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PBH test for block Jordan diagonal case
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Are the following (A, B) pairs C.C.?
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Are the following (A, B) pairs C.C.?
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Observability
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Example
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Observability
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PBH test for diagonal case
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PBH test for block Jordan diagonal case
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Are the following (C, A) pairs C.O.?
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Are the following (C, A) pairs C.O.?
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• Controllability is invariant under transformation

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• Observability invariant under transformation

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Controllability and Observability
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C.C., C.O. and TF poles/zeros
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Intuitive C.C., C.O. and TF
x1
G1(s)
x3
x2
u
G2(s)
G3(s)
C
y
x4
G4(s)
If no pole zero cancellation in each TF and in TF
multiplication, and G1 has poles different from
those of G2, G3, and G4, then C.C. But not C.O.
since x1 and x4 info not available at y
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Intuitive C.C., C.O. and TF
x1
G1(s)
x3
x2
u
G2(s)
3/(s3)
C
y
x4
G4(s)
For the 3/(s3) block, input is x2, state is
x3. One canonical form of state equation for this
1st order TF is given by
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State Feedback
D
1 s
r
u
x



y
B
C


-
A
K
feedback from state x to control u
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Pole placement
Solve this to get ks.
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Example
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• In Matlab
• Given A,B,C,D
• ?Compute QCctrb(A,B)
• ?Check rank(QC)
• If it is n, then
• ?Select any n eigenvalues(must be in complex
  conjugate pairs)
• ev?1 ?2 ?3 ?n
• ?Compute
• Kplace(A,B,ev)

ABk will have eigenvalues at
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Invariance under state feedback

• Thm Controllability is unchanged after state
  feedback.
• But observability may change!

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• Recall linear transformation
• Controllabilitybeing able to use u(t) to drive
  any state to origin in finite time
• Observabilitybeing able to computer any x(0)
  from observed y(t)
• After transformation, eigenvalues, char. poly,
  char. eq, char. values, T.F., poles, zeros are
  unchanged, but eigenvector changed

• Controllability is invariant under transf.

• Observability invariant under transf.

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State Feedback
D
1 s
r
u
x



y
B
C


-
A
K
feedback from state x to control u
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Pole placement

• In Matlab
• Given A,B,C,D
• ?Compute QCctrb(A,B)
• ?Check rank(QC)
• If it is n, then
• ?Select any n eigenvalues(must be in complex
  conjugate pairs)
• ev?1 ?2 ?3 ?n
• ?Compute
• Kplace(A,B,ev)

ABk will have eigenvalues at these values
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Pole placement
Solve this to get ks.