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Peng et al. Intell Robot 2022;2(3):298­312 I http://dx.doi.org/10.20517/ir.2022.27 Page 302

In the following, we will design the decentralized LFC law under the communication network. Similar to [23] ,
a PI controller is used in this paper:
¹
( ) = − ( ) − ( ). (8)
Therefore, dynamic model of the scheduled power systems under the decentralized LFC Equation (8) and
imperfect network environments can be formalized as
(
¤ ( ) = ( ) − ˆ ( ) + ( )
(9)
( ) = ( ), ∈ [ , +1 ),

Í
where ( ) = ( ) = ( ), = [ 1 , · · · , ], = [ , ].
=1
2.2. Impulsive model and study objective
From Equation (5), one can obtain that
(
( ) − ( +1 ), =
( +1 ) = (10)
[ ( ) − ( +1 )] + ( ), ≠
Define an artificial delay ( ) = − , from which one arrives at

≤ , ¤( ) = 1. (11)

0 ≤ ≤ ( ) ≤ +1 − + +1
From Equation (5) and Equation (9), the impulsive power system model can be

Õ
¤ ( ) = ( ) − ( − ( )) − ( ) + ( ). (12)
=1, ≠
ForsystemEquation(12), the initialconditionof ( ) on [− , 0] issupplementedas ( ) = ( ), ∈ [− , 0],
with (0) = 0, where ( ) is a continuous function on [− , 0].

Using scheduling scheme Equation (6) and Equation (7), this paper is to design the decentralized controller
Equation (8) such that system Equation (12) is exponentially stable with a prescribed ∞ performance .

3. MAIN RESULTS
In the following, we first derive sufficient criteria under scheduling scheme Equation (6) and Equation (7) to
ensure the exponential stability of system Equation (12) with a prescribed ∞ performance. Then, criteria are
proposed to design decentralized controllers under multi-channel transmission.

3.1. Stability analysis under the TOD scheduling scheme Eqution (6)
Construct Lyapunov-Krasovskii functional candidate:

Õ
( ) = ( ) ( ) + Π( ) + , (13)

=1
where > 0, > 0, > 0, > 0, > 0, > 0, = 1, · · · , , ∈ [ , +1 ),

¹
Õ
p
2

¤ ( )
,
= 2 ( − )
=1
¹ ¹ ¹
¤ ( ) ¤( ) .
Π( ) = 2 ( − ) 2 ( − )
( ) ( ) + ( ) ( ) +
− −