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Fractional Order Time-Varying-Delay Systems Chapter | 5 139
the delay-independent condition. On the other hand, there is the condition
that takes into account the time-delay length, and it is known as the delay-
dependent condition.
In this section, stability analysis of fractional order time-varying delay
systems will be dealt with. The obtained conditions are very useful in stabil-
ity analysis and controller synthesis. In addition, they will be used in the
next section to design pseudo-state feedback controller law, where the
designed controller law must stabilize the considered unstable fractional
order time-varying delay systems.
The mathematical model for a linear forced fractional order time-varying
delay system can be written as
α
D xðtÞ 5 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞ 1 B u uðtÞ ð5:19aÞ
xðtÞ 5 ψðtÞ; tA½ 2 τ m ; 0; 0 , α , 1 ð5:19bÞ
n
where xðtÞAR is the pseudo-state vector (For an introduction about the
pseudo-state space description see Sabatier et al. (2014)), uðtÞAR m is the
input vector.
Assumption 1: The time-delay function τðtÞ $ 0 is assumed to be continuous,
bounded, and satisfies
0 , τðtÞ , τ m , N ð5:20aÞ
@τðtÞ
0 # # @τ m , 1 ð5:20bÞ
@t
where τ max and @τ max are two constant scalars.
The matrices A 0 , A τ , and B u are known and constant with appropriate
dimensions. The associated function ψðtÞ represents a continuous vector-
valued initial pseudo-states.
5.4.1 Time-Delay-Independent Stability
The purpose of this part is to investigate how the stability of fractional order
time-varying delay systems can be improved by the indirect Lyapunov
method. The authors focus on the case where the time-delay length is not
needed to obtain the delay-independent stability condition.
For analyzing the stability of a linear unforced (without input) fractional
order time-varying delay systems, a special case of the system (Eq. 5.19) is
considered, which can be represented by the following pseudo-state space
equations
α
D xðtÞ 5 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞ ð5:21aÞ
