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The simulation results are given in Figure 5. Figure 5 The synchronized error (pmb{Vert e_{i}(t) Vert _{2}}), (pmb{i=1,2,3,4,5,6}), of system ( 33 ) under pinning control ( 18 ).
An adaptive control scheme is proposed, and the convergence of the synchronized error is guaranteed.
Figure 3 Time series of the synchronized error signals between the master system ( 12 ) and the slave system ( 13 ) in 4D.
The simulation results are given in Figure 9. Figure 9 The synchronized error (pmb{Vert e_{i}(t) Vert _{2}}), (pmb{i=1,ldots,10}), of system ( 32 ) under pinning control ( 3 ).
Unlike directly eliminating the nonlinear dynamics by active control and sliding mode control in the literature, the proposed sliding mode controllers include the equivalent control part, which is only proportional to the synchronized error states, and the switching control part, where the discontinuous control functions have adaptive feedback gains.
As can be seen in Figure 4, the synchronization occurs between the master system (12) and the slave system (24) in 3D. Figure 4 Time series of the synchronized error signals between the master system ( 12 ) and the slave system ( 24 ) in 3D. Figure 1 displays the chaotic attractor of three-dimensional autonomous system (1).
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The synchronizing methods are then compared from various point of views including synchronizing error variance, convergence time, control effort, maximum and minimum values of the control signal.
With nonzero initial conditions, the error system (2.6) is adaptively exponentially synchronized if the synchronization error satisfies (2.7).
The TTS halves the synchronization hops and effectively reduces the multi-hop synchronization error by synchronizing nodes of two adjacent levels.
Yet, in our system model, we assume that the elements of the antenna array are perfectly synchronized or that the synchronization error is small compared to the symbol length.
In contrast to the SES procedure, where all nodes are forced to synchronize in strict time intervals following a pattern (see Figure2), in the HIPESYN scheme any node is able to synchronize to any neighbor node when the synchronization error between both exceeds a fixed threshold, that is, when they really need to synchronize.
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