_{∞}Filter Design for Networked Control Systems: A Markovian Jump System Approach

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This paper puts forward a method to design the _{∞} filter for networked control systems (NCSs) with time delay and data packet loss. Based on the properties of Markovian jump system, the packet loss is treated as a constant probability independent and identically distributed Bernoulli random process. Thus, the stochastic stability condition can be acquired for the filtering error system, which meets an _{∞} performance index level _{∞} performance index level for NCSs with time delay and packet loss can be obtained, which uses linear matrix inequalities (LMIs). Finally, numerical simulation examples demonstrate the effectiveness of the proposed method.

As a new generation of control systems, NCSs [

The Markovian jump system (MJS) refers to a stochastic system with multiple model states and the system transitions between modes in accordance with the properties of the Markov chain due to the multimode transition characteristics of the Markovian jump system in the actual engineering. It can be used to simulate many systems with abrupt characteristics, such as manufacturing systems and fault-tolerant systems [_{2}–_{∞} filter problem of the linear system is explored, and the system has both distributed delay, Markovian jump parameter, and norm bounded parameter uncertainty. In [_{∞} filtering design about a continuous MJS in distributed sampled-data asynchronous is involved; in addition, the system’s mode jumping instants and filter are asynchronous. In [_{∞} filter is studied. However, many conclusions only consider time delay or packet loss separately, which is not very consistent with the actual situation of network application. In addition, the _{∞} filter design of NCSs with time delay and packet dropout has not yet been considered widely.

Based on this, this paper studies the stability of networked control systems considering both time delay and packet loss. Although the analysis process is more complicated than considering time delay or packet loss alone, however the conclusion is more general and universal, and then the existence conditions of system filters are given. The effectiveness of the proposed method is verified by simulation, and the relevant conclusions are more practical. In Section _{∞} performance of system is proven. In Section _{∞} filter for NCSs with time delay and packet loss is designed. In Section

Notation: here are some of the symbols in the paper. The superscript “_{x}-dimensional Euclidean space.

First, consider the following kind of discrete-time linear systems [_{2}[0, ∞), which indicates the measured output; _{i} with appropriate dimensions. _{i}, _{i}, _{i}, and _{i} also have the same situation.

After calculations, another important expectation can be shown as follows:

For network packet loss, both Bernoulli distribution and Poisson distribution have been considered. According to the network protocols adopted in actual systems, such as industrial Ethernet and profibus, it is more practical to model network packet loss with Bernoulli distribution in this paper.

Mathematical model description of filtering is in the following formula:_{f i}, _{f i}, _{f i}, and _{f i} are real matrices to be determined with compatible dimensions. Combining (

Obviously, a filter error system (

(see [

(see [_{∞} performance index level γ under zero conditions for all nonzero

Based on the previous known conditions, a stochastically stable condition meets an _{∞} performance index level

If there exists symmetric matrices _{i} and _{i} > 0, consider NCSs with filter in (

Consider a Lyapunov functional candidate as follows:

Applying Schur complement theorem to the above

From Theorem _{min} (-Λ) indicates the minimum value of the eigenvalue for -Λ and _{min} (-Λ)} is the lower bound of _{min} (-Λ); for any

Thus, we can get

As a result, it can be proved that system (

For NCSs (_{i}, _{i} > 0 satisfied the following matrix inequalities, and the system (_{∞} norm performance level

When

By Schur complement, (_{∞} norm performance level

The proof is over.

Here, we will go to solve the system filter.

Consider NCSs (_{1i} > 0, _{3i} > 0, _{1i} > 0, _{3i} > 0, _{i} > 0, and _{i} > 0 and _{2i}, _{2i}, _{i}, _{Fi}, _{Fi}, _{Fi}, and _{Fi} satisfied the following matrix inequalities, then

System (_{∞} norm performance level

Slack matrix approach can be used for (

Then, we have following equation using (

According to equations in (

The proof is over.

Consider system (

Two models in the simulation are set as those data in (_{∞} norm performance level

Parameters change of

Parameters change of

Parameters change of different

Parameters change of filter parameters.

Parameters change of output parameters.

Figure

Figure

Figure

Figure

Figure

As can be seen from the ordinate of the system state diagram of Figure

This paper has investigated the _{∞} filtering problem about NCSs with time delay and packet dropout phenomenon. Packet dropout is treated as a constant probability independent and identically distributed Bernoulli random process. By the Lyapunov stability theory, system (_{∞} norm performance level _{∞} performance index level of NCSs with time delay and packet dropout is completed. Finally, a simulation result is given to prove the validity of the new design scheme.

The data used to support the findings of this study are included within the article. Because it is a numerical simulation example, readers can get the same results as this article by using the LMI toolbox of Matlab and the theorem given in this article.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant nos. 61403278 and 61503280).

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