wilamowski-b-m-irwin-j-d-industrial-communication-systems-2011

23-6 Industrial Communication Systems
23.2.3 Impulsive Control for Chaos Synchronization via Partial States
In the continuous synchronization control strategies like the linear feedback control and adaptive
control above, continuous transmission of state variables of the driving system is required to obtain
synchronization control signals in implementation. Thus, the security level is significantly decreased if
the scheme is applied to secure communication. On the other hand, a purely impulsive synchronization
scheme is implemented by transferring the states information of the driving system only at discrete
impulsive instants, therefore, is much more feasible in secure communication to enhance transmission
security and also to improve the transmission efficiency. In addition, a secure communication system
based on impulsive synchronization has another advantage of being less sensitive to channel noise than
that based on continuous synchronization.
Suppose that the driving system is given by (23.1). The response system with impulsive control added
to partial states Ỹ of the system at only certain discrete-time instants τ l is given as
⎧X̃̇ = − AX̃ + f ( X̃ , Ỹ
) + h1( t),
⎪
⎪
⎨Y ̃̇ = g( X̃ , Ỹ ) + h2( t), t ≠ τl, l = 0, 1, 2, …,
⎪
⎪Ỹ
( τ Ỹ
l + ) = ( τl − ) + Ul( t), t = τl,
⎩⎪
(23.7)
where {τ l } is a discrete-time set satisfying that 0 < τ 0 < τ 1 < … < τ l < τ l+1 < …, and τ l → ∞ as l → ∞
and τ l is called an impulsive instant. The design of impulsive controller involves the choices of the
control law U l (t) and impulsive instant τ l . According to [WWSG09], U l (t) is chosen in such a way that
E ( τ + ) = . This gives
2 l 0
−
Ul( t) = −E2( τ l ) .
(23.8)
With such a control, ∆ ̃ ̃ +
Y Y Y ̃ −
= ( τ ) − ( τ ) ≠ 0, and thus there is a “jump” of the state vector Ỹ at τ l.
t = τl
l
l
To ensure that system (23.1) and (23.7) are asymptotically synchronized with the control in (23.8), the
impulsive instant τ l should be chosen from the following strategy [WWSG09].
Strategy for the choice of impulsive instant τ l
1. If the states e n+j (t), j = 1, …, m, are monotonic for t ∈ ( τ
+ −
l , τl+
1 ], l = 0, 1, 2, …, then the l + 1th impulsive
control effect is added at the moment of τ l+1 .
2. If one of the state errors e n+j (t), j = 1, …, m, is not monotonic for t ∈ ( τ
+ −
l , τl+
1 ], l = 0, 1, 2, …, then the
l + 1th impulsive control effect is added at the time instant τ′ l + 1 at which e n+j (t) changes monotonicity,
i.e., τl+ 1 = τ′
l+
1.
To implement the strategy using digital device, the monotonic condition is approximated in programming.
During the implementation, an iterative step size T is required by applying Runge–Kutta
method to solve the state equations. The strategy of choosing the impulsive instant τ l , l > 0, is illustrated
in the flowchart shown in Figure 23.5 [WWSG09], where an upper bound τ of impulsive intervals is
determined according to the conditions given in [WWSG09].
The same example as shown in the feedback control section is now considered. In the implementation,
the upper lound τ is chosen as 0.005 based on the given strategy. The synchronization errors of
two identical Chua’s circuits under impulsive control are shown in Figure 23.6.
Again, the errors converge to zero asymptotically.
© 2011 by Taylor and Francis Group, LLC