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Edited by: Yanzheng Zhu, Western Sydney University, Australia

Reviewed by: Bo Zhao, Beijing Normal University, China; Tiao-Yang Cai, Hebei Normal University, China; Junyi Wang, Loughborough University, United Kingdom

This article was submitted to Autonomic Neuroscience, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this study, the stability for a class of sampled-data Takagi-Sugeno (T-S) fuzzy systems with state quantization was investigated. Using the discontinuous Lyapunov-Krasoskii functional (LKF) approach and the free-matrix-based integral inequality bounds processing technique, a stability condition with less conservativeness has been obtained, and the controller of the sampled-data T-S fuzzy system with the quantized state has been designed. Furthermore, based on the results, the sampled-data T-S fuzzy system without the state quantization was also discussed, and the required controller constructed. The results of two simulation examples show that both the maximum sampling intervals, with and without the quantized state for T-S fuzzy systems, are actually superior to the existing results.

In the real world, most physical systems and processes can be modeled mathematically as complex nonlinear systems. Because some parts of nonlinear systems are always coupled and influence each other, it is difficult to analyze and synthesize these systems. Therefore, establishing an effective and suitable control model to address this issue in nonlinear systems, is significant. In recent years, T-S fuzzy systems has been an effective method to analyze and synthesize nonlinear systems, because of the fact that the T-S fuzzy model is able to transform a complex nonlinear system into several simple linear systems with membership functions, approximating the nonlinear function smoothly with an arbitrary precision in the closed set space, which is ubiquitous in chemical processes, robotics systems, and automatic systems (Tanaka and Wang,

At the front line of other research, there has been increasing interest in the sampled-data control system, a rapid development of digital and communication technology (Chen and Francis,

It is well-known that due to the limited capacity and energy consumption in the network system, it is especially important to quantize the sampled-data before transmission. However, most existing studies assume that the data transmission can be performed with infinite precision, and the impact of quantization is always ignored in the network environment. On the other hand, sampling quantization before signal transmission may lead to a limited cycle and chaos. Thus, the study of sampled-data systems with state quantization has attracted significant research attention (Fu and Xie,

Motivated by that, this work mainly discusses the stability and stabilization control problem for nonlinear T-S fuzzy sampled-data systems with quantized states. By using the discontinuous LKF approach and free-matrix-based integral inequality boundary processing technique, stability conditions with less conservativeness are obtained for T-S fuzzy systems, with sampled-data and quantized states, and the controllers are designed accordingly. Furthermore, the stability of T-S fuzzy sampled-data systems without quantized states is also analyzed utilizing the above theoretical results, and the required sampling data controllers are designed simultaneously. The main contributions of this paper can be summarized as follows: (1) In constructing LKF aspects, it is not necessary that the discontinuous LKF approach applied in this work are positive for all time _{k} and _{k+1}, which can broaden the restriction in LKF. (2) In estimating the derivation of LKF, the free-matrix-based integral inequality boundary processing technique is used to provide more freedom in deriving stability for sampled-date T-S fuzzy systems. From two points of view, the conservativeness of stability conditions can be decreased for sampled-data T-S fuzzy systems with and without quantized states effectively through the methods of our design in this work.

^{T}.

A class of continuous-time nonlinear systems can be described using the following T-S fuzzy model

The rules of plant _{1}(_{i1}(_{n}(_{in}(

where _{i} and _{i} are any matrices with appropriate dimensions. θ(_{1}(_{n}(_{ij}(

where _{i}(θ(_{ij}(θ_{j}(_{j}(_{ij}, which has the following properties. For θ(_{i}(θ(_{i}(θ(

Suppose the control signal is a sequence of holding times generated by a zero-order-holder function.

This paper designs the controller based on the state feedback for T-S fuzzy systems described in (1) and uses the idea of parallel distributed compensation to share the same fuzzy set with the fuzzy model in the premise part of the designed fuzzy controller.

Controller rules _{1}(_{k}) is _{j1}(_{n}(_{k}) is _{jn}(

where _{j} is the state feedback gain matrix with appropriate dimension, and _{k}) is the discrete measurement value at the sampling time of _{k}.

The logarithm quantizer is described as

The _{m}(•) is symmetric, and therefore we have

The quantizer satisfies the following quantization criteria

where ρ_{m} and _{m}(•) is strictly defined as follows

where _{m} = 1 − ρ_{m}/1 + ρ_{m}(_{m}(_{k}) > 0, the following relationship is established

For _{m}(_{k}) < 0, the following is satisfied

Therefore, the quantizer can be expressed as

where

Hence, the overall state feedback controller with the quantized state can be designed

Suppose the distance between two consecutive sampling instants belongs to an interval, and then for all

where _{L} and _{U} are known constants satisfying 0 ≤ _{L} ≤ _{U}. Combining Equations (1)–(6), the T-S fuzzy sampled-data system with state quantization can be obtained as follows

The following lemma is important for further analysis.

_{1},

where

_{k})ϕ(_{k},

In this section, the asymptotic stability conditions of T-S fuzzy systems with state quantization are analyzed.

For the convenience of system analysis and design, we define

where _{i} is defined as the block entry matrix. Other notations are defined as

_{L} and _{U}, satisfying 0 ≤ _{L} ≤ _{U}, system (Equation 7) is asymptotically stable, if there are some symmetric positive definite matrices _{1}, _{1}, _{2}, _{2}, _{1}, _{2} with appropriate dimensions, and for any

where

with

and _{1}, _{2} have been defined in Lemma 1.

Proof: The novel discontinuous LKF in this work is constructed as follows

where

with

_{3}(_{1}(_{2}(_{k} and _{k+1}. All this can decrease the conservativeness of stability conditions effectively.

The derivative of _{1}(_{2}(_{3}(

By Lemma 1, we obtain the inequality

Based on the closed-loop system (Equation 7), the following equality holds

and it can be further written as

From Equation (4), for the diagonal matrix

Then combining (Equations 11–16), an upper bound of

where

with the following two equalities

Therefore, using Schur Complement, Equations (8) and (9) are equivalent to

Additionally, Theorem 1 has provided the stability results for T-S fuzzy sampled-data systems (Equation 7) with state quantization, and, the following Theorem 2 will be given in order to obtain the sampled-data controller.

_{L} and _{U}, satisfying 0 ≤ _{L} ≤ _{U}, system (Equation 7) is asymptotically stable, if there are some symmetric positive definite matrices _{j},

where

with

The gain matrix _{j} of the sampled-data controller with state quantization is defined as

Proof: Define

Equation (8) is pre- and post-multiplied by

_{k})) in _{2}(

In this section, the impact of the quantized state is not considered, and the T-S fuzzy sampled-data system then becomes

Define

Now, using the same method used in Theorem 1, we have the following Corollary without considering the state quantization.

_{L} and _{U} with 0 ≤ _{L} ≤ _{U}, the system (Equation 20) is asymptotically stable, if there are some symmetric positive definite matrices _{1}, _{1}, _{2}, _{2}, _{1}, _{2} with appropriate dimensions, and for any

where

with

where _{1}, _{2}, _{1} and ℜ_{2} have been defined in Theorem 1.

Furthermore, the following Corollary regarding the sampled-data controllers design can be derived using a similar method used in Theorem 2.

_{L} and _{U} with 0 ≤ _{L} ≤ _{U}, system (Equation 20) is asymptotically stable, if there exist symmetric positive definite matrices _{j},

where

with

where _{j} of the sampled-data controller without state quantization is defined as

This section provides two numerical examples to demonstrate the effectiveness and superiority of the proposed method.

The Lorenz system (Equation 25) can be represented as a type of T-S fuzzy system (Equation 2) with the following parameters

and the membership functions satisfy _{2}(_{1}(_{1}(_{1}(

Here we choose

Case I: For the case with the quantized state, take the quantizer densities as

and the quantizer parameter is supposed as

Considering the quantized state based on Theorem 2, when ε = 0.1, the allowable maximum sampling period that can ensure the asymptotic stability of system (Equation 1) is 0.0742, which is larger than 0.0503 obtained in Liu et al. (

The response curves of system (Equation 25) under the initial condition ^{T} with the obtained gain matrices are given in

State responses of system (Equation 25) with state quantization in Example 1.

Control input of system (Equation 25) with state quantization in Example 1.

Case II: For the case without state quantization, based on Corollary 2, as ε = 0.1, the allowable maximum sampling period ensuring the asymptotic stability of system (Equation 7) is 0.0741. And the allowable upper bounds

Maximum allowable bounds

Lam et al., |
Zhu et al., |
Wu et al., |
Liu et al., |
Corollary 2 | |

0.0158 | 0.0270 | 0.0347 | 0.0560 | 0.0741 |

Simulation results are provided to verify the effectiveness of the proposed method. When

The response curves of the system (Equation 25) with the initial condition ^{T} under the obtained gain matrices are given in

State responses of system (Equation 25) without state quantization in Example 1.

Control input of system (Equation 25) without state quantization in Example 1.

Note that the unified chaotic system (Equation 26) with _{1}(

and the membership functions are _{1}(_{1}(_{1}(_{2}(_{1}(_{1}(

Here

Case I: Taking the same quantizer densities as in Example 1, and using Theorem 2, when ε = 0.01, system (Equation 26) can be asymptotically stable with the maximum sampling period

Under the obtained gain matrices and the initial condition ^{T}, the state response and control input

State responses of system (Equation 26) with state quantization in Example 2.

Control input of system (Equation 26) with state quantization in Example 2.

Case II: The maximum allowable upper bound of the sampling interval

Maximum allowable bounds

Zhu et al., |
Wu et al., |
Liu et al., |
Corollary 2 | |

0.0377 | 0.0480 | 0.0830 | 0.1040 |

When ε = 0.1 and

The response curves of system (Equation 26) with initial condition ^{T} under the above gain matrices are displayed in

State responses of system (Equation 26) without state quantization in Example 2.

Control input of system (Equation 26) without state quantization in Example 2.

In this work, we have investigated the stability for a class of nonlinear T-S fuzzy sampled-data systems with state quantification. A new LKF approach has been constructed and a Free-Matrix-Based boundary treatment technique for integral inequalities has been adopted in order to obtain less conservative stability conditions and correspondingly, a controller has been designed. Furthermore, the stability of the T-S fuzzy sampled-data system without quantized states, has also been discussed and sampled-data controllers have been designed accordingly. The experimental results show that the maximum sampling interval for T-S fuzzy sampled-data systems with and without quantized states in our work, are both larger than the results in previous studies. Nevertheless, some other interesting problems that need to be addressed still exist, such as the reliable control design for the sampled-data T-S fuzzy systems with state quantization, and the extension of our developed approaches to the dissipativity-based sampled-data control design, which deserve further investigation.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the Natural Science Foundation of China under Grant 61503045, 61603121, National Natural Science Foundation of Jilin Province under Grant 20180101333JC.

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