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Page 10 of 23 Tan et al. Complex Eng Syst 2023;3:6 I http://dx.doi.org/10.20517/ces.2023.10

= 1 ( ( ), )

1 ∑ T
= lim {[ { + = | = } ( + )P ( + )]

→0
+
=1, ≠
T T
+ [ { + = | = } ( + )P ( + ) − ( )P ( )]}

1 ∑ { + = , = } T
= lim [ ( + )P ( + )]

→0 { = }
+
=1, ≠
1 { + = , = } T T
+ lim [ ( + )P ( + ) − ( )P ( )]

→0 { = }
+

1 ∑ ( (ℎ + ) − (ℎ)) T (21)
= lim [ ( + )P ( + )]

→0 1 − (ℎ)
+
=1, ≠
1 1 − (ℎ + ) T T
+ lim [ ( + )P ( + ) − ( )P ( )]

→0 1 − (ℎ)
+

1 ∑ ( (ℎ + ) − (ℎ)) T
= lim [ ( + )P ( + )]

→0 1 − (ℎ)
+
=1, ≠
1 1 − (ℎ + ) T T
+ lim [ ( ( + ) − ( ))P ( + )]

→0 1 − (ℎ)
+
1 1 − (ℎ + ) T (ℎ + ) − (ℎ) T
+ lim [ ( )P ( ( + ) − ( )) − ( )P ( )],

→0 1 − (ℎ) 1 − (ℎ)
+
where ℎ is the dwell time when the system jumps from the previous mode to mode , (ℎ) represents
the cumulative distribution function of residence time when the system (10) maintains in th mode,
represents the probability density from mode to mode . Using the properties of cumulative distribution
function, it can be seen that
1 − (ℎ + ) 1 (ℎ + ) − (ℎ)
lim = 1, lim = (ℎ), (22)
+
→0 + 1 − (ℎ) →0 1 − (ℎ)
where (ℎ) represents the transition probability of the system in mode . When ≠ , we have (ℎ) =
∑ T ∑
(ℎ) and (ℎ) = − =1, ≠ (ℎ), one derives that = 1 ( ( ), ) = ( )( =1, ≠ (ℎ)P ) ( )

+ 2 ( )P ¤ ( ).
T

According to [22] , there is a scalar −1 ∈ (0, ], and we know that
−1
1 0

∑ ∑ ∑ ∑ ∑
T T T −1 T
2 ( )P G ( ) ≤ ( 1 ( )G P G ( ) + ( )P ( )). (23)

1
=1 =1, ≠ =1 =1, ≠ =1, ≠
Similarly, the = 2 ( ( ), ), = 3 ( ( ), ), = 4 ( ( ), ) can be written as
T T
= 2 ( ( ), ) = ( )Q ( ) − ( − )Q ( − )

∫ ∑
T T
+ ( )Q ( ) + ( )(−Q + (ℎ)Q ) ( ) ,

−
=1
∫
2 T T

= 3 ( ( ), ) = ¤ ( )R ¤ ( ) − ¤ ( )R ¤ ( ) ,

−

1 ∑ ( ) T ( ) ∑ ( ) T ( )
= 4 ( ( ), ) = (Δ ( )) Ω Δ ( ) − 0 (Δ ( )) Ω Δ ( )
( )
=1 =1

∑
T ( ) T ( )
≤ ( − ( ))Ω ( − ( )) − 0 ( ( )) Ω ( ).

=1