1

 LFT: Definition
 Properties
 Examples
 General Technique
 HIMAT Example
 Redheffer Star Product

2

 Let M be a partitioned matrix
 A (lower) linear fractional transformation (LFT) of M over D is defined as
 F_{l}(M, D):=M_{11}+M_{12}
D(IM_{22}
D)^{–1}M_{21}
 _{ }where D
has suitable dimensions and IM_{22} D is invertible.
 Similarly, an (upper) LFT is defined as
 F_{u}(M, D_{u}):=M_{22}+M_{21 }D_{u} (IM_{11}
D_{u})^{–1}M_{12}

3

 F_{l}(M, D)
is wellposed if IM_{22} D is invertible.
 (F_{u}(M, D_{u})
)^{–1 }= F_{u}(N, D_{u}) with N given by
 Suppose C is invertible. Then
 (A+BQ)(C+DQ)^{–1}=
F_{l}(M, Q),
(C+DQ)^{–1 }(A+BQ)= F_{l}(N, Q)
 If M_{12} is invertible, then F_{l}(M, Q)=(C+DQ)^{–1
}(A+BQ)
 with A=M_{12}^{–1}M_{11 }, B=M_{21}  M_{22 }M_{12}^{–1}M_{11
}, C=M_{12}^{–1}, and D=M_{22 }M_{12}^{–1}
 If M_{21} is invertible,
then F_{l}(M,
Q)=(A+BQ)(C+DQ)^{–1}
 with A=M_{11 }M_{21}^{–1}, B=M_{12} – M_{11 }M_{21}^{–1}M_{22
}, C=M_{21}^{–1}, and D=_{ }M_{21}^{–1
}M_{22}

4

 Feedback System: consider a feedback system with disturbance d, sensor
noise n, we can write this system in the LFT form with external inputs (d,
n) and controlled outputs (v,u_{f}) such that z = F_{l}(G,K)w with

5

 Here we show more steps for putting in the LFT form. Note that G is the
transfer matrix between (w,u) and (z,y). It does not include the
controller K. It is better to ``pull out’’ the K in the first place as
shown below.
 Now find the G as the transfer
matrix between (w,u) and (z,y):

6

 A direct state space realization for G can also be obtained by
connecting the state space realizations of the components. Let
 That is,
 Now define a new state vector x=(x_{p}, x_{f},x_{u},
x_{v}) and eliminate the variable y_{p} to get a
realization of G as

7


8

 Uncertain Function: assume each coefficient has some perturbation.

9

 More steps: pulling out the deltas.

10

 Parametric UncertaintyA Mass/Spring/Damper System: assume each
coefficient has some perturbation.

11

 HIMAT Example: Consider a HIMAT system with
 The openloop interconnection matrix is

12

 The interconnection matrix:

13


14


15

 Let P and K be two matrices with appropriate dimensions.
 Then the following interconnection, called a star product, is also a
matrix, denoted by P*K

16

 Let P and K be two transfer matrices with state space realizations. Then
the transfer matrix P*K is given by
 Matlab
 >> P*K =
starp(P,K,dimy,dimu)
 >> F_{l}(P,K) = starp(P,K)
