1.2. UNCERTAINTY MODELS 3
a and b can be written in the same way as a
0
C ı
a
and b
0
C ı
b
, respectively. a
0
D
1
L
0
C
0
and b
0
D
R
0
L
0
show the nominal values of a and b, respectively. According to the values of R
0
,
L
0
, C
0
, ı
R
, ı
L
, and ı
C
, a and b can be written as:
a
min
< a < a
max
b
min
< b < b
max
:
(1.2)
So, Equation (1.1) no longer describes a single transfer function. It is a family of transfer func-
tions with uncertain coefficients. e term, “Interval Plant Model” is used for such plants (i.e.,
a plant with uncertain coefficients). Lower/upper bound of coefficients variations is known.
1.2.2 UNSTRUCTURED UNCERTAINTY
Many dynamic perturbations that may occur in different parts of a system can, however, be
lumped into one single perturbation block , for instance, some unmodeled, high-frequency
dynamics. is uncertainty representation is referred to as “unstructured” uncertainty. In the
case of linear, time-invariant systems, the block may be represented by an unknown transfer
function matrix. e unstructured dynamics uncertainty in a control system can be described
in different ways [Gu et al., 2013]. e most famouse ones are additive and input/output mul-
tiplicative perturbation configurations. If G
p
.
s
/
; G
o
.
s
/
, and I show the perturbed system dy-
namics, a nominal model description of the physical system and identity matrix, respectively,
then:
• additive perturbation: G
p
.
s
/
D G
o
.
s
/
C .s/,
• input multiplicative perturbation: G
p
.
s
/
D G
o
.
s
/
ŒI C
.
s
/
, and
• output multiplicative perturbation: G
p
.
s
/
D ŒI C
.
s
/
G
o
.
s
/
.
In Single Input Single Output (SISO) systems, there is no difference between input mul-
tiplicative perturbation and output multiplicative perturbation. In Multi Input Multi Output
(MIMO) systems the two descriptions are not necessarily the same.
1.2.3 STRUCTURED UNCERTAINTY
In some problems the uncertain parts can be taken out from the dynamics and the whole sys-
tem can be rearranged in a standard configuration of (upper) Linear Fractional Transformation
F .M; /. e uncertian block would then have the following general form:
D diag
˚
ı
1
I
r
1
; ı
2
I
r
2
; ı
3
I
r
3
; : : : ; ı
s
I
r
s
;
1
; : : : ;
f
; ı
i
2 C;
j
2 C
m
j
m
j
;
where
P
s
iD1
r
i
C
P
f
j D1
m
j
D n with n is the dimension of the block . So, consist of s re-
peated scalar blocks and f full blocks. e full blocks need not be square. Since the considered
has a certain structure, such description is called structured.
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