Abstract
Consider the linear discrete-time fractional order systems with uncertainty on the initial state
{
Δ
α
x
i
+
1
=
A
x
i
+
B
u
i
,
i
≥
0
x
0
=
τ
0
+
τ
⌢
0
∈
ℝ
n
,
τ
⌢
0
∈
Ω
,
y
i
=
C
x
i
,
i
≥
0
\left\{ {\matrix{{{\Delta ^\alpha }{x_{i + 1}} = A{x_i} + B{u_i},} \hfill & {i \ge 0} \hfill \cr {{x_0} = {\tau _0} + {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0} \in {\mathbb{R}^n},} \hfill & {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0} \in \Omega ,} \hfill \cr {{y_i} = C{x_{i,}}\,\,\,i \ge 0} \hfill & {} \hfill \cr } } \right.
where A, B and C are appropriate matrices, x0 is the initial state, yi is the signal output, α the order of the derivative, τ0 and
τ
⌢
0
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}
are the known and unknown part of x0, respectively, ui = Kxi is feedback control and Ω ⊂ ℝn is a polytope convex of vertices w1, w2, . . . , wp. According to the Krein–Milman theorem, we suppose that
τ
⌢
0
=
∑
j
=
1
p
α
j
w
j
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0} = \sum\limits_{j = 1}^p {{\alpha _j}{w_j}}
for some unknown coefficients α1 ≥ 0, . . . , αp ≥ 0 such that
∑
j
=
1
p
α
j
=
1
\sum\limits_{j = 1}^p {{\alpha _j} = 1}
. In this paper, the fractional derivative is defined in the Grünwald–Letnikov sense. We investigate the characterisation of the set χ(
τ
⌢
0
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}
, ϵ) of all possible gain matrix K that makes the system insensitive to the unknown part
τ
⌢
0
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}
, which means
χ
(
τ
⌢
0
,
∈
)
=
{
K
∈
ℝ
m
×
n
/
‖
∂
y
i
∂
α
j
‖
≤
∈
,
∀
j
=
1
,
…
,
p
,
∀
i
≥
0
}
\chi \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } }_0}, \in } \right) = \left\{ {K \in {\mathbb{R}^{m \times n}}/\left\| {{{\partial {y_i}} \over {\partial {\alpha _j}}}} \right\| \le \in ,\forall j = 1, \ldots ,p,\,\forall i \ge 0} \right\}
, where the inequality
‖
∂
y
i
∂
α
j
‖
≤
∈
\left\| {{{\partial {y_i}} \over {\partial {\alpha _j}}}} \right\| \le \in
showing the sensitivity of yi relatively to uncertainties
{
α
j
}
j
=
1
p
\left\{ {{\alpha _j}} \right\}_{j = 1}^p
will not achieve the specified threshold ϵ > 0. We establish, under certain hypothesis, the finite determination of χ(
τ
⌢
0
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \tau } _0}
, ϵ) and we propose an algorithmic approach to made explicit characterisation of such set.
Application of Magnetic and Ultrasonic Methods for Determining Parameters of Ferromagnetic Components in Iron Ore Slurry Flows
