Page 149 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 137

Here, a useful inequality is given, which will be used later in the proof of

Γ 11 Γ 12
the main results. For any semipositive definite matrix Γ 5 Γ 22 $ 0,
the inequality
ð t
T
T
τ max η ðtÞΓηðtÞ 2 η ðtÞΓηðtÞds $ 0 ð5:11Þ
t2τðtÞ
T
T T
holds, where ηðtÞ 5 x ðtÞ x ðt2τðtÞÞ .
5.3 PROBLEM FORMULATION

Firstly, the following fractional order linear time-invariant systems with a
constant time-delay of retarded type is considered
α
D xðtÞ 5 A 0 xðtÞ 1 A τ xðt 2 τÞ ð5:12aÞ
xðtÞ 5 ψðtÞ; tA½ 2 τ; 0Š; 0 , α , 1 ð5:12bÞ
n
where xðtÞAR is the pseudo-state vector (for an introduction about the
1
pseudo-state space description see Sabatier et al. (2014)). τAR is a constant
time-delay. The matrices A 0 ; A τ AR n 3 n are known and constant. The associ-
ated function ψðtÞ represents a continuous vector-valued initial pseudo-state.
The characteristic equation of the system (5.12), which is quasi-
α 2sτ
polynomial in s and e , is given by
α 2sτ
detðs I 2 A 2 A τ e Þ 5 0 ð5:13Þ
Generally, the characteristic Eq. (5.13) can be rewritten as follows
K
α
α
X
Mðs; τÞ 5 pðs Þ 1 q i ðs Þe 2isτ ð5:14Þ
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α
α
where τAR 1 is the delay parameter, pðs Þ and q i ðs Þ for i 5 1; 2:::; K are
α
polynomials functions of s and deg p $ deg q i . The considered polynomials
α
degrees are the degrees in the variable s , then they are integers.
Lemma 2: (Fioravanti et al., 2012) Let G be a strictly proper system with
characteristic equation given by (5.14) satisfying deg p $ deg q i for
i 5 1; 2; .. .; K being thus of retarded type. Then G is BIBO-stable if and only if
G has no poles in ℜðsÞ $ 0 (in particular, no poles of fractional order at s 5 0).
It is shown in Fioravanti et al. (2012), that by applying a variable substi-
α
tution as ς 5 s , a practical test for stability can achieved. Using this substi-
tution, the characteristic Eq. (5.14) becomes
K
X 1
Mðς; τÞ 5 pðςÞ 1 q i ðςÞe 2iς ατ ð5:15Þ
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