1. Introduction
1. | (1)To alleviate the impact of FDIA on the performance of the MJNNs, SSDC is developed to investigate the security control of the controlled system; |
2. | (2)The model of the error system with multi-delays is established. Associated with it, a sampling-period-probability-dependent LLF is constructed to reduce the conservatism of the main results. |
3. | (3)A less conservative exponential synchronization criterion and a design algorithm for determining the controller gain are given, respectively. |
2. Problem description
Figure 1. The structure of the secure control under FDIAs. |
Suppose there exist m sampling periods satisfying $0={h}^{0}\lt {h}^{1}\lt {h}^{2}\lt \cdots \lt {h}^{m}$, and the occurrence probability of each sampling period is recognized as $\mathrm{Prob}\{{h}_{k}={h}^{j}\}={\beta }_{j},\,j=1,2,\ldots ,m$. Obviously, ${\sum }_{j=1}^{m}{\beta }_{j}=1.$
For g(t), suppose there exists constant matrix G satisfying
Note that different from the deterministic SDC, SSDC has multiple sampling periods, which can reflect the inherent characteristics of the sampling error. Hence, the SSDC system can not only ensure the system achieves the desired performance in a nonideal environment, but also reduce the redundancy of communication transmission. In light of the characteristics of SSDC, the position of dj(t) can be described by introducing a series of random variables ${\gamma }_{j}(t)$, which follow the Bernoulli distribution.
In an unsatisfactory network environment, attackers tamper with the transmission data by launching FDIAs to destroy system performance. In view of the randomness of the cyber attack, the data transmitted to the controller can be given by $\varepsilon (t)$ with random variable $\alpha (t)$. Under the FDIAs, SSDC can alleviate the impact of the attacks on the performance of the controlled system by adjusting the occurrence probability of the sampling periods.
3. Main results
For given scalars $\alpha \gt 0,\delta \gt 0,{h}^{j}\,\gt 0,{\beta }_{j}\gt 0$, and feedback gain K, the master-slave MJNNs (
The time-dependent LLF is constructed as follows:
In this paper, the time-dependent looped-functional ${ \mathcal W }(t)$ is constructed. From ${ \mathcal W }(t)$, we can easily see that the positive definiteness of the functional matrices ${U}_{j},{\overline{V}}_{j},{X}_{j}$ are not required. That is, ${ \mathcal W }(t)$ relaxes the positive definite constraint of ${ \mathcal W }(t)$ in the sampling interval, and only requires its positive definiteness at the sampling instants $\{{t}_{k}\}$, it contributes to relaxing the conservativeness of the stability conditions and to increasing the maximum allowable bound of the sampling intervals.
For given constants $\alpha \gt 0,\delta \gt 0,{h}^{j}\gt 0,{\beta }_{j}\gt 0,\lambda $, in mean square sense, the master-slave MJNNs (
In view of the LMIs (
Significantly, the ${\beta }_{j}(t)$-dependent LLF ${ \mathcal W }(t)$ contains the information of occurrence probability ${\beta }_{j}$ for different sampling periods, thus, the derived synchronization conditions are ${\beta }_{j}$-dependent. When facing cyber attacks, we can adjust ${\beta }_{j}$ to obtain an appropriate control gain K.
4. Numerical example
Table 1. The maximum value of h2 for different h1 with β1 = 0.6, β2 = 0.4, δ = 0.5. |
h1 | 0.01 | 0.05 | 0.1 | 0.15 | 0.2 |
---|---|---|---|---|---|
${h}_{\max }^{2}$ with α = 0.2 | 0.834 | 0.786 | 0.682 | 0.533 | 0.363 |
${h}_{\max }^{2}$ with α = 0.5 | 0.457 | 0.426 | 0.357 | 0.258 |
Table 2. The maximum value of h2 for different β1 with δ = 0.1, h1 = 0.1. |
β1 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|
${h}_{\max }^{2}$ | 0.304 | 0.330 | 0.367 | 0.420 | 0.509 | 0.682 | 1.113 | 2.637 |
Figure 2. (a) The time response of r(t) and ϵ(t); (b) Markov jumping mode; (c) the control input; (d) the attack signal g(x(t)) and α(t). |