Improved stabilization criteria for fuzzy systems under variable sampling

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Abstract

This paper investigates the problem of stabilization for fuzzy sampled-data systems with variable sampling. A novel Lyapunov–Krasovskii functional (LKF) is introduced to the fuzzy systems. The benefit of the new approach is that the LKF develops more information about actual sampling pattern of the fuzzy sampled-data systems. In addition, some symmetric matrices involved in the LKF are not required to be positive definite. Based on a recently introduced Wirtinger-based integral inequality that has been shown to be less conservative than Jensen’s inequality, much less conservative stabilization conditions are obtained. Then, the corresponding sampled-data controller can be synthesized by solving a set of linear matrix inequalities (LMIs). Finally, an illustrative example is given to show the feasibility and effectiveness of the proposed method.

Introduction

In the past several years, considerable attention has been attracted to Takagi–Sugeno (T–S) fuzzy system [1] from both the academic and industrial communities, and many important results have been reported in this field [2], [3], [4], [5]. A nonlinear dynamical system can be denoted as linear models through T–S fuzzy model. By using fuzzy membership functions, the total fuzzy model can be acquired by compounding the linear models. Then, a discrete gain-scheduled state feedback controller is designed for linear model. The total controller is a fuzzy mixture of each local linear controller. To an extent, the T–S fuzzy model method can be used to facilitate the analysis and synthesis a nonlinear dynamic system [6], [7], [8], [9], [10].

With the rapid development of digital computing science, the digital devices which have the advantage of low installation cost, better reliability and easy maintenance are gradually utilized in industrial applications. This allows fuzzy system only using the samples of continuous-time measurement signals at discrete time instants. These samples are used to control the continuous-time plant through a zero-order hold (ZOH). The approach drastically increases the efficiency of bandwidth usage and reduces the system information. This control system is considered to be sampled-data system. Because the control signals between any two continuous sampling instants will be held constant and only be changed at each sampling time, the analysis and synthesis of sampled-date systems are difficult and complex. The most popular method which has been widely used to sampled-data system is the input delay method [11], [12], [13], [14], [15]. Based on this method, the sampled-data system is described as a continuous-time system with time-varying delay generated by the ZOH. Then, the stability conditions in terms of LMIs will be established by the LKF method. Under constant sampling, in [16], network-based robust passive control for fuzzy systems with randomly occurring uncertainties was investigated and in [17], [18], [19], the T–S fuzzy sampled-data control for chaotic systems was discussed. Stability analysis result for sampled-data polynomial-fuzzy-model-based control systems was developed in [20]. As is well known, in digital control system uniform sampling is a commonly used method. However, due to the uncertainty of the system itself and external disturbances, the uniform sampling period cannot be always guaranteed. Thus, it is necessary to study the case of time-varying sampling intervals.

For the time-varying sampling case, in [21], the problem of sampled-data H filtering was investigated for a continuous-time T–S fuzzy system with an interval time-varying delay. A new LKF was proposed to take more information about the relationship between the current state and its delayed state into account. Recently, the distributed sampled-data asynchronous H filtering problem was investigated in [22] for a continuous-time Markovian jump linear systems. The simultaneous presence of variable sampling, network-induced delays and Markovian switching sensing topologies was in the systems. For an actual system, the authors in [23] investigated the problem of H controller design for a class of uncertain vehicle suspension systems with sampling measurements. A Lyapunov functional method was investigated to establish the H performance, and the controller design was converted into a convex optimization problem with LMI constrains. In [24] the problem of vehicle active suspension control with frequency band constraints and actuator input delay was considered. Based on the Kalman–Yakubovich–Popov lemma, the finite-frequency problems were converted into a set of LMIs to be solved. For the problem of robust H sampled-data control, the authors proposed a robust H sampled-data state feedback controller for the uncertain fuzzy systems with nonuniform sampling in [25]. Recently, [26] investigated sampled-data robust H control for T–S fuzzy systems with time delay and uncertainties, where the free weighting matrices were introduce to deal with the integral items. Based on this method, the coupling time-varying matrix inequalities were converted into a set of decoupling matrix inequalities in order to obtain the innovative delay-dependent stabilization condition. In [27], the problem of stabilization for sampled-data fuzzy systems was investigated under variable sampling. Based on an appropriate enlargement scheme, an improved input delay method was proposed. However, it is worth pointing out that the method of membership functions is not considered in the aforesaid literature, which may be conservative. In addition, network-based tracking control for the T–S fuzzy system was investigated in [28] based on the deviation bounds of asynchronous normalized membership functions. Some less conservative conditions were established because of the constraints on membership functions. By applying deviation bounds of asynchronous normalized membership functions, the network-based output tracking control for a T–S fuzzy system using an event-triggered communication scheme was investigated in [29]. The numerical example illustrated that using this approach can improve the tracking control performance. Therefore, it is important and necessary to further study the the membership function derivations to design and analyze the sampled-data control fuzzy systems, which is the motivation in this study.

Based on above discussion, we will deal with the problem of stabilization for sampled-data fuzzy systems with variable sampling. By use of the looped LKF approach and Wirtinger-based integral inequality, the stabilization conditions are derived in the term of LMIs such that the fuzzy systems are stable. Then, by applying a lemma on the membership function deviations, the corresponding sampled-data controller can be synthesized to obtain the maximum of sampling interval. It should be pointed out that the LKF develops more information on the actual sampling pattern about the sampled-data fuzzy systems. Some less conservative criteria are obtained such that the closed-loop systems is asymptotically stable.

In summary, the main contributions of this paper are: (1) The sampled-data control stabilization for T–S fuzzy systems by using the membership function deviations is investigated; (2) A novel looped LKF that makes full use of the information on the actual sampling pattern is constructed; (3) Some less conservative stabilization analysis for chaotic system is obtained based on the fuzzy sampled-data control.

Notations: Throughout this paper, a real symmetric matrix P > 0(≥ 0) denotes that P is a positive definite (positive semi-definite) matrix, and X > Y(X ≥ Y) means XY>0(0). Rn is the n-dimensional Euclidean space. Rm×n denotes the set of m × n real matrix. I is the identity matrix. For any positive integer r, Zr:={1,2,,r}. The transpose of a matrix A is denoted by AT. Moreover, for any square matrix ARn×n, we define He(A)=A+AT. The set of symmetric matrices of dimension n is denoted by Sn, and a symmetric matrix by [ABCD]=[AB*D].

Section snippets

Problem formulation

The dynamic of chaotic system can be described as follows: x˙(t)=f(x(t),u(t)),where x(t)Rn is the vector of system state, u(t)Rm is the vector of control input and f( · ) is a nonlinear function that satisfies f(0,0)=0. By using the method of sector nonlinearity proposed in [30], the chaotic system (1) can be described as the following T–S fuzzy system:

Plant rule i: IF θ1(t) is ηi1, θ2(t) is ηi2 and … and θp(t) is ηip, THEN x˙(t)=Aix(t)+Biu(t),iZr,where r is the number of fuzzy rules, ηi1,ηi2

Stability analysis

In this section, we discuss the stabilization problems of closed-loop fuzzy system (8) by defining a looped LKF and using the Wirtinger-based integral inequality. For the simplicity of representation, the following notations are used: ei=[0n×(i1)nIn×n0n×(4i)n],i=1,2,3,4,Υ1=e1e2,Υ2=e1+e22e4.

Theorem 1

Consider the closed-loop fuzzy sampled-data system (8) with the membership functions satisfying Eq. (10), and suppose the gain matricesKj(jZr) of the sampled-data controllers (5) are given. For given

Controller design

In this section, the sampled-data controller design method for the fuzzy system (3) is proposed by Theorem 1, and a sufficient criteria for the existence of the appropriate sampled-data controller is present as follows.

Theorem 2

Consider the closed-loop fuzzy system (8) and sample-data controller (5) with the membership function satisfying (10). For given scalarsh>0,δs>0(sZr),ɛ1,ɛ2, if there exist n × n matricesP¯>0,R¯4>0,R¯i(i=1,2,3)Sn,T¯,X¯1,X¯2,m×n matricesK¯j and 4n × n matricesM¯1ij,M¯2ij,4n×4n

Numerical examples

In this section, an numerical example is given to show the effectiveness of the proposed approach.

Example 1

Consider the following dynamic equation: {x.1(t)=x2(t)x.2(t)=gsin(x1(t))amlx22(t)sin(2x1(t))2acos(x1(t))u(t)4l/3amlcos2(x1(t)),where x1(t) denotes the angle displacement of the pendulum, x2(t) denotes the angular velocity of the pendulum, g=9.8m·s2 is the acceleration of gravity, m is the mass of the pendulum, M is the mass of the cart, 2l is the length of the pendulum and u(t) is the force

Conclusion

In this paper, a novel augmented LKF has been introduced to analysis the stability of the fuzzy systems with sampled-data control. The benefit of the new method is that the LKF takes into account more information about actual sampling pattern of the fuzzy sampled-data systems. The conditions have been obtained using Wirtinger’s inequality which encompasses the Jensen’s inequality to deal with the cross terms. The approach of membership function deviations which establish the quantitative

Acknowledgment

This paper is partially supported by the National Natural Science Foundation of China (61503120), the Natural Science Foundation of Hebei Province (F2016209382), and the Fostering Talents Foundation of North China University of Science and Technology (JP201511). Also, the work of J.H. Park was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number NRF-2017R1A2B2004671).

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