1. Introduction
Since the 1980s, neural networks have been playing a key role in more and more areas such as image compression, signal processing, artificial intelligence, and so on [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18]. However, as application demand increases, the usual neural networks based on lower algebra inevitably show their weaknesses or even powerlessness in some complicated and advanced fields. Fortunately, the application areas of neural networks develop rapidly because of the progress made by higher algebra, and concepts such as complex or quaternion are introduced to construct USOMDVNN [
4,
8,
9,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43].
In the real synchronous control system, the signal transmission sometimes needs to depend on the general communication network. For example, these include secure communication in wireless networks, robot cooperation in harsh environments, and multi-agent distributed formation control. The system of neural networks based on multi-dimension is employed to the networked synchronous control system for modeling and analysis. More applications can be realized through the remote control and remote operation of the control system. In addition, more and more researchers have employed the synchronization of complex dynamic networks in information science, power systems, and biological systems, and so on [
2,
3,
4,
5,
6,
7,
17,
18]. Both the structure and function of the brain’s nervous system can be simulated by more complex dynamic networks with multi-dimension at different levels. Meanwhile, some researchers have applied more complex dynamic networks with multi-dimension in planning the robotic path, collaboratively controlling unmanned aerial vehicles, and searching satellite formation. Naturally, there is an innovative idea for combining neural networks with higher algebra to build complex neural networks that can simultaneously have more in-depth improvement.
Recently, some researchers have focused on various dynamics of neural networks with real algebra [
13,
16,
44], neural networks in the area of the complex field [
8,
32,
33], or neural networks in the quaternion field [
4,
9,
19,
20,
21,
22,
23,
24,
25,
26]. Among the various dynamical behaviors, stability is the basic one that receives close attention from more scholars [
4,
19,
20]. Particularly, owing to the definition of master–slave synchronization, the research of the master system’s synchronization with the slave one is equal to the discussion about the stability of the related error system with the master–slave systems.Therefore, many researchers have investigated the different kinds of synchronization criteria on the various systems of USOMDVNN [
22,
23,
24,
25,
26]. Asymptotic synchronization and exponential synchronization are both considered as infinite-time synchronization, whose essence indicates that synchronization can just be finished, as long as the time verges to infinity [
19,
20,
21,
29,
30,
31]. Different with the above two kinds of synchronization, finite-time synchronization [
23] owns absolute advantages in practical engineering applications. For finite-time synchronization, synchronization could be realized in a certain amount of time, which depends on the initial value. However, the certain amount of time can not be estimated without the accurate initial value. In 2012, Polyakov, in [
11], put forward the thought of fixed-time synchronization, which indicates that the master systems can synchronize with the slave ones in a finite time without the precise initial value. That is, the finite time has the fixed upper bound for the discussed system under any initial conditions. Very recently, on one hand, many researchers have devoted themselves to deriving the fixed time with the tighter upper bound, by establishing some formulas [
12,
13,
45]. On the other hand, because the basic commutative law can not be directly applied in the quaternion multiplication, many researchers have focused on studying different methods such as separation or nonseparation in solving the unavoidable difficulty [
4,
5,
19,
20]. The research about different dynamics in the system of USOMDVNN with many methods is still meaningful and worthy of much attention, as Liu et al. in [
19] has said. Focusing on the previous results on the special systems of RVNN [
13,
16,
44], CVNN [
8,
29,
35,
36], and QVNN [
21,
22,
23], it is not difficult to find out that some analyses, especially those based on the separation of the systems of CVNN and QVNN, may inevitably increase the amount of calculation. Therefore, we need the search for the concise LKF and switching controllers for the novel synchronization criteria, including both asymptotic synchronization and the fixed-time one about the unified system of USOMDVNN.
Motivated by the above investigation and research, we put forward a unified analysis on globally asymptotic synchronization and fixed-time synchronization for USOMDVNN in this paper. First of all, the general model of USOMDVNN is successfully constructed, mainly on the basis of multidimensional algebra, Kirchhoff current law, and neuronal feature. Then, the new inequalities with some parameters are applied in a unified form whose variable can be translated into its special one, such as real, complex, and quaternion. It is worth mentioning that the useful parameters really make some contributions to the novel structure of the concise LKF, the new construction of the switching controllers, and the easy acquisition of the flexible criteria. Further, we acquire the newer criteria mainly by employing Lyapunov analysis, constructing new LKF, applying two unified inequalities, and designing novel controllers. It is worth mentioning that the concrete value of the fixed time is less than the other ones in the existing results, owing to the adjustable parameters. The following points are the the main contributions of this paper.
(1) The unified model of the investigated USOMDVNN is generally set up.
(2) Owing to the new establishment of the extended derivative of the absolute value function and the new application of the generalized Cauchy–Schwarz inequality, the concise LKF is successfully constructed and is used to solve the problem of non-commutativity for multidimensional multiplication.
(3) Inspired by the previous work [
25,
26], both the newly constructed LKF and the derived criteria contain the flexible parameters
, and the novelly designed controllers involve the adjustable parameters
. By adjusting the values of
properly, the LKF can become multiple, the relevant conditions both in Theorems 1 and 2 can become flexible, and the controllers in Theorem 2 can reach the smaller synchronization fixed time
.
Notations: The sets of all
-dimensional matrices that own multidimensional algebra are denoted as
. Define the multi-dimension-valued variable
as follows.
Correspondingly, the derivative of variable
in the multi-dimension-valued form is regarded to be multi-dimension formed by the derivatives of every element
(
M can be equal to be
or
K;
M can be equal to be
R or
I; or
M can be equal to be
R) of the multi-dimension variable
with respect to
t:
More specific descriptions can be found in the previous works [
25,
26].
2. Model Description and Preliminaries
First of all, we give the general master system of USOMDVNN as follows.
where
,
,
stands for the state variable of the
m-th neuron;
,
(
) describes the decay matrix; the external input vector is denoted as
; the corresponding multi-dimension-valued activation functions are expressed by
.
The relevant slave system of USOMDVNN can be similarly established as follows,
where,
;
stands for the state variable of the
m-th neuron;
,
(
) describes the corresponding decay matrix;
denotes the subsequently designed controllers.
Generally, design the switching controllers as follows,
where,
,
,
,
,
,
;
stand for the gains of the above controllers, and all can be any nonzero constants.
Denote
. According to the above master system and slave one, we can obtain the following general error one:
where,
,
.
Definition 1 ([
19])
. The error system of USOMDVNN (3) is defined to achieve the globally asymptotical stability if . Definition 2 ([
23])
. The error system of USOMDVNN (3) is considered to achieve the finite-time stability if and for , where, is a constant and . Moreover, the finite setting time is denoted as . Definition 3 ([
13])
. The error system of USOMDVNN (3) is regarded to achieve the fixed-time stability, if the following conditions hold: (1) The error system of USOMDVNN (3) can achieve the finite-time stability; (2) Let is a positive and fixed constant. Then, there exists , for any . Moreover, is defined to be the fixed time. Assumption 1. Assume that the activation functions could satisfy the basic inequality , where , .
Lemma 1 ([
11])
. Suppose to be a radially continuous and unbounded function, and satisfy: (1) ⟺; (2) Any solution of the error system of USOMDVNN (3) meets , for , , and , where stands for the upper right-hand Dini differential of . Then, the error system of USOMDVNN (3) can achieve the fixed-time stability and the concrete fixed time . Lemma 2 ([
27] (Extended Cauchy-Schwarz Inequality))
. For any , , where, , () are all constants. Lemma 3 ([
26] (Extended Differential of Absolute Value Function))
. If , , then , , where the symbol · describes the dot product between the two vectors. Lemma 4 ([
28])
. If , , , then , . 4. Numerical Simulation
In this part, three numerical simulations will be supported to demonstrate the above theoretical results and compare the criteria with some existing results.
Example 1. The following master system of QVNN with 2-neuron is considered.where, , , , , , , , , , , . The corresponding slave system of QVNN with 2-neuronwhere, , , , , , , , and the switching controllers are desined in (4) where, , , , , , . Let . Because the value of varies, the acquired criteria are diverse and not just one. Here, the following two conditions of the values of and are discussed, and the final values of fixed time are listed in Table 1 according to Theorems 1 and 2, respectively: (1) Condition 1: , ,
(2) Condition 2: , .
In addition, we simulate the corresponding state change diagrams of the different systems of USOMDVNN (18) and (18) in 30 random initial values from Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. (From Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, and denote and , respectively.) From Figure 1, Figure 2, Figure 3 and Figure 4, the divided momentary error phase diagrams for the four elements between for the variable of (18) and of (19) in 30 random initial values for globally asymptotical synchronization are presented. From Figure 5, Figure 6, Figure 7 and Figure 8, the divided momentary error phase diagrams for the four elements between for the variable of (18) and of (19) in 30 random initial values for the fixed-time synchronization are presented. From Figure 9, Figure 10, Figure 11 and Figure 12, the mixed momentary error phase diagrams for the four elements between for the variable of (18) and of (19) in 30 random initial values for the above two kinds of synchronization are presented.
Figure 1.
Instantaneous error states for the R part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Figure 1.
Instantaneous error states for the R part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Figure 2.
Instantaneous error states for the I part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Figure 2.
Instantaneous error states for the I part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Table 1.
The related values under two cases.
Table 1.
The related values under two cases.
Condition | | |
---|
1 | −16.37 | 2.4142 |
2 | −15.185 | 0.8321 |
Remark 7. In this example, we just give two choices about the values of . As the value of changes, the final criteria vary accordingly. Obviously, the two conditions are satisfied, and the two criteria have been derived and the different values of have been acquired. Moreover, by comparison with the results in Table 1, we can judge that the value of under condition 2 is less than the one under condition 1. This indicates that the parameter has really played an important part in adjusting the final results. That is, as we adjust the values of properly, the final value of can be less and less, in order to satisfy the application need of engineering.
Figure 3.
Instantaneous error states for the J part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Figure 3.
Instantaneous error states for the J part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Remark 8. From Figure 1, Figure 2, Figure 3 and Figure 4, we show the state trajectories of the error system under linear controllers, and the globally asymptotical synchronization has been realized. From Figure 5, Figure 6, Figure 7 and Figure 8, we demonstrate that the state trajectories of the error system under the nonlinear controllers and the fixed-time synchronization have been achieved. Moreover, we can find that the final time for asymptotical synchronization is more than the one for fixed-time synchronization. This indicates that the nonlinear controllers are both novel and effective. In general, the designed controller (4) is a newly switching one that not only plays an important part in the common synchronization problem, but also has an effect on the popular fixed-time one. Example 2. The following 2-neuron CVNN is considered to be the same as in Ding et al. [8],where , , , , , , , , , , , , , , , The corresponding 2-neuron slave system of CVNN,where , , , , , , , , , , , , , , , and the switching controllers are desined in (4), where, , , , , , . Let and , respectively. Because of the changed values of , the acquired criteria are diverse, and not just one. Here, the following two conditions of the values of the and are discussed for Theorems 1 and 2, respectively:
(1) Condition 1: , , .
(2) Condition 2: , , .
By simple calculation, we obtain the corresponding values under the above two cases in Table 2 and Table 3. It is easily seen that we acquire the corresponding criteria and compare the fixed time with the results in [8] under different conditions.
Figure 4.
Instantaneous error states for the K part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Figure 4.
Instantaneous error states for the K part between for the variable
of (
18) and
of (
19) under 30 random initial values for asymptotical synchronization.
Figure 5.
Instantaneous error states for the R part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Figure 5.
Instantaneous error states for the R part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Example 3. Consider the same 2-neuron RVNN, as in Chen et al. [13],where , , , , , , , , , , , . The corresponding 2-neuron slave system of RVNN:where , , , , , , , , , , , and the switching controllers are desined in (4), where, , , , , , . Let and , respectively. Due to the different values of , the acquired criteria are diverse and not just one. Here, the following two conditions of the values of and are discussed about the conditions in Theorems 1 and 2, respectively:
(1) Condition 1: , , .
(2) Condition 2: , , .
By simple calculation, we obtain the corresponding values under the above two cases in Table 4 and Table 5. It is easily seen that we acquire the corresponding criteria and compare the fixed time with other ones from the above tables.
Figure 6.
Instantaneous error states for the I part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Figure 6.
Instantaneous error states for the I part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Table 2.
The related values under two cases.
Table 2.
The related values under two cases.
Condition | | |
---|
1 | −6.525 | 3.9142 |
2 | −4.7814 | 1.1808 |
Table 3.
The comparison of the different fixed time .
Table 3.
The comparison of the different fixed time .
Method | |
---|
[8] | 4.5 |
condition 1 | 3.9142 |
condition 2 | 1.1808 |
Figure 7.
Instantaneous error states for the J part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Figure 7.
Instantaneous error states for the J part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Table 4.
The related values under two cases.
Table 4.
The related values under two cases.
Condition | | |
---|
1 | −0.675 | 2.4142 |
2 | −1.675 | 0.8321 |
Table 5.
The comparison of the different fixed time .
Table 5.
The comparison of the different fixed time .
Method | |
---|
[11] | 4.3784 |
[45] | 4.3620 |
[12] | 3.4259 |
[13] | 2.3897 |
condition 1 | 2.4142 |
condition 2 | 0.8321 |
Figure 8.
Instantaneous error states for the K part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Figure 8.
Instantaneous error states for the K part between for the variable
of (
18) and
of (
19) under 30 random initial values for fixed-time synchronization.
Figure 9.
Instantaneous error states for the R part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Figure 9.
Instantaneous error states for the R part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Remark 9. From the above three examples, our acquired results have made the following progress by the comparison with the ones in [11,12,13,45]: (1) The established model of USOMDVNN (1) can be reduced the special cases in [11,12,13,45]; (2) The solution to discuss the related dynamics of different multi-dimension neural networks can be unified in this paper; (3) The controllers are switching and can be various, owing to the participation of the parameters ; (4) Whether the criteria for asymptotical synchronization or for a fixed-time one, the final judgements are newer; (5) The final values of the fixed time are easy to be calculated; (6) The listed values of in Table 5 are less than the ones in [11,12,13,45].
Figure 10.
Instantaneous error states for the I part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Figure 10.
Instantaneous error states for the I part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Figure 11.
Instantaneous error states for the J part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Figure 11.
Instantaneous error states for the J part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Figure 12.
Instantaneous error states for the K part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.
Figure 12.
Instantaneous error states for the K part between for the variable
of (
18) and
of (
19) under 30 random initial values for two kinds of synchronization.