In addition, based on the same finite-time stability theory and the same sliding mode control, we ensure that the trajectory of error system converges to a chosen sliding surface within finite time and remains on it forever.
The time derivative of V along the trajectory of error system (5) is V ˙ = e T e ˙ + e θ T e ˙ θ + e β T e ˙ β = e T e ˙ + e θ T ( − θ ˙ r ) + e β T ( − β ˙ r ).
For example, Wiersema and colleagues [ 6] examined the developmental trajectory of error processing in children (aged 7-8), young adolescent 13-144), adultsults (age 23-24).
Figure 5 State trajectories of errors (pmb{e_{1} k)}).
In addition, an appropriate sliding mode controller is designed to drive the state trajectories of error system to the prescribed sliding surface in finite time and remain on it evermore.
Figure 5 The trajectories of error signals (pmb{e_{i1}}), (pmb{e_{i2}}), (pmb{e_{i3}}) ( (pmb{i=1,2,3,4}) ) for system in Example 4.3 with pinning impulsive control.
Figure 2 The trajectories of error signals (pmb{e_{11}}), (pmb{e_{12}}), (pmb{e_{21}}), (pmb{e_{22}}) for system in Example 4.1 with pinning impulsive control.
Figure 3 The trajectories of error signals (pmb{e_{i1}}), (pmb{e_{i2}}) ( (pmb{i=1,2,3,4}) ) for system in Example 4.2 with pinning impulsive control.
Under the action of adaptive controller (4), taking the initial conditions as x ( 0 ) = ( 1.2, − 0.8 ) T and y ( 0 ) = ( − 12, 8 ) T, α = β = 0.1, we show the state trajectories of error system (2) and updated laws in adaptive controller (4) in Figures 2-4, respectively.
In other words, an integral sliding mode controller is designed such that the sliding motion is globally asymptotically stable, and the state trajectory of the error system (4) is globally driven onto the specified sliding surface and maintained there for all subsequent time.
The designed tracking controller can not only suppress the mismatched perturbations when the controlled dynamics (master slave) are in the sliding mode, but also drive the trajectories of synchronization errors into a small bounded region whose size can be adjusted through the designed parameters.
