Research paper
Networked sampled-data control of distributed parameter systems via distributed sensor networks

https://doi.org/10.1016/j.cnsns.2021.105773Get rights and content

Highlights

  • This paper studies the problem of the distributed networked sampled data control for distributed parameter systems.

  • The distributed sensor network is proposed to obtain the sampled data measurements of the state.

  • A distributed sampled data output feedback controller based on the distributed sensor network is designed.

  • The global exponential stability criteria are obtained under three different boundary conditions, respectively.

Abstract

This paper studies the problem of the distributed networked sampled-data control for a class of distributed parameter systems with spatially-dependent diffusion term. In view of limited fixed sampling spatial points, the distributed sensor network is proposed to obtain the sampled-data measurements of the state which can efficaciously circumvent blind sampling area or sampling error. A distributed sampled data output feedback controller based on the distributed sensor network is designed to ensure the stabilization of the distributed parameter systems with the time delays induced by the communication network. Based on Lyapunov method, linear matrix inequality technique and time-delay approach, the global exponential stability criteria are obtained for the closed-loop distributed parameter systems under three different boundary conditions, respectively. Finally, the numerical simulation proves the effectiveness of the controller, and numerical comparison shows that the proposed control method is less conservative.

Introduction

With the advanced development of the digital sensor in the last few decades, sampled-data control has attracted special attention of numerous researchers due to its advantages, such as high accuracy and reliability, effective interference suppression, and good versatility. The design of sampled-data control scheme plays the crucial role in the applications of digital implementation issues as well as theoretical development. Consequently, a great deal of studies have been focused on the sampled-data control problem for various kinds of finite dimensional systems such as the works [1], [2], [3], [4].

Notice the fact that a large body of real systems including heat transfer process and reaction-diffusion process cannot be described by lumped parameter systems with finite dimensions, an increasing interest is given to the sampled-data control of distributed parameter systems (DPSs) with infinite dimensions. The study of sampled-data control problems of DPSs is becoming active and has become a hot spot in the control field. Recently, Tarn et al. proposed a stabilization scheme in [5] by using periodic output feedback control instead of continuously monitoring. Based on a generalized sampled method with a weighting function, Logemann et al. in [6] derived the exponential stability of the infinite-dimensional system. In [7], Logemann et al. constructed a sampled-data feedback controller to stabilize the infinite-dimensional system and a series of exponentially stable conditions were presented by using semigroup theory. By employing piecewise polynomial controls, Rebarber and Townley in [8] obtained some necessary and sufficient conditions for stabilization of continuous-time DPSs. Moreover, Logemann in [9] presented the necessary and sufficient condition for the existence of the stabilizing linear sampled-data controller. Recently, Cheng et al. in [10] proposed a sampled-data strategy for a boundary control problem of a class of DPSs. These sampled controllers seem to be easy to implement in that the controlled objects are all linear time-invariant systems. For nonlinear infinite-dimensional systems, Wang and Wu in [11] studied the sampled-data fuzzy control problem. Moreover, for a class of nonlinear DPSs with sampled-data measurements, Ghantasala and El-Farra in [12] considered the active fault-tolerant control issue, where the model reduction method (e.g., Galerkin’s technique) was suggested to obtain a finite-dimensional system which can represent the main characteristics of the studied DPSs, and the finite-dimensional controller was designed based on the finite-dimensional system. However, it seems to be a huge challenge to achieve an exact performance of the original DPSs because of the truncation before controller design.

To avoid this drawback, the design proposals of infinite-dimensional controllers were proposed recently to obtain better control performance, see the works [13], [14], [15], [16], [17] and the references therein. Selivanov and Fridman in [13] used the time-delay approach to study the sampled-data relay control problem for semilinear parabolic system. Kang and Fridman in [14] employed the time-delay approach and Lyapunov-Krasovskii method to deal with the sampled-data control problem and to derive the regional exponential stability conditions. Most recently, Wang et al. in [15], [16] considered to combine Lyapunov-Krasovskii method and Takagi-Sugeno fuzzy model approach to derive the sampled-data exponential stability conditions. However, the results in [13] were of regional or semi-global stabilization, and the approach in [14], [15] was inapplicable to the DPSs with spatially-dependent diffusion coefficients. To solve the global stability problems for DPSs with spatially-dependent diffusion coefficients and uncertain nonlinear terms, Fridman and Blighovsky in [17] presented a robust sampled-data controller and derived the decay rate of the exponential convergence. However, the problem of allocating resources in a distributed fashion is ignored in the research [17], where the sensors collect information just from a single resource, which may poses the blind sampling area or sampling error. Then, to make full use of sensor network resources effectively is an advantageous approach to obtain the precise sampled data of the systems.

Motivated by the above discussion, in this paper, we study the network-based sampled-data control of a class of DPSs with spatially-dependent diffusion term. In view of limited fixed sampling spatial points, the distributed sensor network is proposed to obtain the sampled-data measurements of the state which is consisting of groups of sensors is considered in the sampling processing, and the sensors provide a series of state measurements to their own sensor group by the interaction among sensor nodes based on the prescribed sensing topology, which can efficaciously circumvent blind sampling area or sampling error. Then, a networked sampled-data static output feedback controller is proposed to stable DPSs with the time delays induced by the communication network. The global exponential stability criteria in terms of LMIs are obtained for DPSs with three different boundary conditions, respectively. Further, the upper bounds on the sampling intervals and the upper bounds on the resulting decay rate are obtained in this paper. Finally, a numerical example under three different boundary conditions is given to illustrate the effectiveness of the obtained results.

The main contribution and advantage of this paper can be summarized as follows:

  • (1)

    A distributed sensor networked is employed to obtain more precise sampled data of the systems, which can efficaciously circumvent blind sampling area or sampling error caused by a single sensor, where M sensors are considered to provide the measurement for spatial sampled point in each sub-domains by collaborative interaction information among sensors according to a prescribed sensing topology, which yields improved environment perception and system efficiency while providing desired information;

  • (2)

    Different from the finite-dimensional sampled data controllers in [12], the static output feedback controller proposed in this paper is an infinite-dimensional sampled data controller based on the output measurement of the infinite-dimensional distributed parameter systems without any information lost, which can accurately and effectively stable the DPSs with spatially-dependent diffusion term. Compared with the impulsive controller in [18], which requires a large number of spatial point sampling measurements, the networked sampled-data controller based on a finite number of point output measurements in this paper is easy and convenient to implement;

  • (3)

    Compared with the stability criteria in [17], [19], the stability criteria in this paper make better control performance of the system and relax the restrictions on the system by using our controller and control method, which can be shown in numerical example.

Notations: In this paper, the set of all real numbers is denoted by R; The set of n-dimensional vectors is denoted by Rn; Rm×n stands for the set of all real m×n matrices; I represents the appropriate dimension identity matrix; In represents the n×n identity matrix; * stands for the symmetric matrix; exp means exponent; Let H=L2([0,l];R) be a Sobolev space which is produced by square integrable absolute functions M(ϑ,t)Rn, ϑ[0,l]R, t0; C1 stands for a set of smooth functions; M(ϑ,t) stands for the partial derivative of M(ϑ,t) with respect to ϑ.

Section snippets

Plant: diffusion semilinear DPSs

Consider a spatially-distributed processes model, which dynamic trajectories are governed by the following DPSsM(ϑ,t)t=ΔD(ϑ)M(ϑ,t)+EM(ϑ,t)+AM(ϑ,t)+f(M(ϑ,t),t)+u(ϑ,t),where M(ϑ,t)=[M1(ϑ,t),,Mn(ϑ,t)]TRn is state variable; ϑ[0,l],t0; the diffusion term is considered asΔD(ϑ)M(ϑ,t)=[ϑ(d1(ϑ)Mϑ(ϑ,t)),,ϑ(dn(ϑ)Mϑ(ϑ,t))],is the diffusion term with di(ϑ)C1 such that 0<di0<di(ϑ) for ϑ[0,l],i=1,2,,n; f(M(ϑ,t))=[f1(M1(ϑ,t)),,fn(Mn(ϑ,t))]T (f:Rn×[0,+)Rn) is considered as continuous and

Main results

Before the main results are given, we first introduce the following assumption and useful lemmas.

Assumption 1

There exists constant scalars fmi,fMi such thatfmifi(M(ϑ,t),t)fi(M^(ϑ,t),t)M(ϑ,t)M^(ϑ,t)fMi,holds for any M(ϑ,t),M^(ϑ,t)Rn, where f(·,0)=0,M(ϑ,t)M^(ϑ,t).

Lemma 1

[26]

Let a<b, S=ST>0, thenabω˙T(s)Rω˙(s)1baΥTdiag{S,3S,5S}Υ,whereΥ=[Υ1T,Υ2T,Υ3T]T,Υ1=ω(b)ω(a),Υ2=ω(b)+ω(a)2baabω(s)ds,Υ3=ω(b)ω(a)12(ba)2ab(sb+a2)ω(s)ds.

Lemma 2

[27]

If f1,f2,,fN:RmR have positive value in an open set DRm, then the reciprocating

Numerical example

In order to illustrate effectiveness of our results, the following example is given.

Consider the distributed parameter system as followsM(ϑ,t)t=ΔD(ϑ)M(ϑ,t)+EM(ϑ,t)+AM(ϑ,t)+f(M(ϑ,t),t)+i=1Nχi(ϑ)ui(ϑ,t),where the domain of the space is [0,π], which is divided into 10 sub-domains on average. For each sub-domain, there are 4 sensors to provide the measurement of the sampled point. The parameters of the systems (46) are given as: ui(ϑ,t)=ΛiM(ϑ¯i,tk), D(ϑ)=0.5, E=0.1, A=0.3, Λi=Ki(DI)Ci=0.1, I=

Conclusion

In this paper, the distributed networked control problem is investigated for a class of DPSs governed by semilinear diffusion PDEs. The distributed sensor network is considered to provide the precise measurements, and a distributed networked sampled-data controller is designed to ensure the stabilization of the distributed parameter systems. Moreover, the time delays induced by the communication of the network is considered. To facilitate analysis of the closed-loop system with sampled-data,

CRediT authorship contribution statement

Huihui Ji: Project administration, Validation, Formal analysis, Validation. Baotong Cui: Project administration, Writing - review & editing. Xinzhi Liu: Formal analysis, Writing - review & editing.

Declaration of Competing Interest

The authors declare that there are no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. This work was supported in part supported by the NSERC.

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