LMI-Based Sliding Mode Robust Control For A Class Of Multi-Agent Linear .

Open Access Library Journal2022, Volume 9, e8342ISSN Online: 2333-9721ISSN Print: 2333-9705LMI-Based Sliding Mode Robust Control for aClass of Multi-Agent Linear SystemsTongxing Li, Wenyi Wang, Yongfeng Zhang, Xiaoyu TanSchool of Mathematics and Statistics, Taishan University, Taian, ChinaHow to cite this paper: Li, T.X., Wang,W.Y., Zhang, Y.F. and Tan, X.Y. (2022)LMI-Based Sliding Mode Robust Controlfor a Class of Multi-Agent Linear Systems.Open Access Library Journal, 9: d: December 30, 2021Accepted: January 18, 2022Published: January 21, 2022Copyright 2022 by author(s) and OpenAccess Library Inc.This work is licensed under the CreativeCommons Attribution InternationalLicense (CC BY en AccessAbstractThis study deals with the multi-agent linear system, which is a more realisticand accurate discrete model with disturbance terms. Based on linear matrixinequality technology and sliding mode control, we give the forward-feedbackcontrol term. Furthermore, sufficient conditions for the closed-loop systemare established by Lyapunov stability theory. Results of simulation show thatthe proposed method is effective.Subject AreasAutomataKeywordsLinear Matrix Inequality, Multi-Agent Linear System, Asymptotic Stability1. IntroductionVariable structure control (VSC) is essentially a special kind of non-linear control, the discontinuity of control is a prominent feature of its nonlinearity. Thecontrol strategy of VSC is to force the system to move according to the state trajectory of the predetermined sliding mode according to the current state of thesystem in the dynamic process. So VSC is also called sliding mode control(SMC).VSC appeared in the 1950s and has been developed and perfected for morethan 50 years. Now it has formed a relatively independent research branch andbecome a general design method of automatic control system. It is suitable forlinear and non-linear systems, continuous and discrete systems, deterministicand uncertain systems, centralized and distributed parameter systems, centralized and decentralized control, etc. And it has been gradually applied in practicDOI: 10.4236/oalib.1108342Jan. 21, 20221Open Access Library Journal

T. X. Li et al.al engineering, such as motor and power system control, robot control, aircraftcontrol, satellite attitude control and so on [1] [2] [3] [4] [5]. This control method makes the system state slide along the sliding surface by switching the control variables, and makes the system invariable when subject to parameter perturbation and external disturbance. Because of this characteristic, VSC methodhas attracted wide attention of scholars all over the world [6] [7] [8]. Variablestructure systems (VSS) and its main mode of operation SMC are recognized asone of the most efficient tools to deal with uncertain systems due to their robustness and even insensitivity to perturbations [9] [10] [11].In [2] a terminal SMC strategy with projection operator adaptive law is proposed in a hybrid energy storage system (HESS). The controller can be designedby the constraint condition, combining the projection operator adaptive law.Linear matrix inequality (LMI) is a powerful design tool in the field of control.Many control theory and analysis and synthesis problems can be simplified tocorresponding LMI problems, which can be solved by constructing effective computer algorithms. LMI technology has become an effective tool in control engineering, system identification, structural design and other fields. Using LMItechnology to solve some control problems is an important direction in the development of control theory. Recently, the problem of LMI-based sliding moderobust control has received significant attention due to its important applications [12] [13].A LMI based sliding surface design method for integral sliding-mode controlof mismatched uncertain systems has been presented in [14]. Moreover, [15] investigated the chaos control problem for a general class of chaotic systems basedon SMC via LMI. The robust stability of uncertain linear neutral systems withtime-varying discrete and distributed delays is investigated via a descriptor modeltransformation and the decomposition technique of the discrete-delay term matrix. In the form of a LMI, they put forward delay-dependent stability criteria in[4]. A SMC method via LMI was presented in [5], which is used for the fluttersuppression problem in supersonic airflow.The optimal sampled-data state feedback control for continuous-time Markovjump linear systems (MJLS) was designed in [16]. Stability and performance robustness against polytopic uncertainty acting on the system parameters including the transition rate matrix are analyzed through differential linear matrix inequalities [16].Based on LMI technology and SMC, we first consider the multi-agent linearsystem with disturbance terms to our best knowledge. The main contributionsand primary distinctions of this paper with other works can be given as follows.1) A more realistic and accurate discrete model with disturbance terms is proposed which is relevant for many practical sampled data systems;2) New forms for forward-feedback control term and sliding mode robustterm are proposed in this paper;3) Sufficient conditions for the closed-loop system are established using Lyapunov stability theory.DOI: 10.4236/oalib.11083422Open Access Library Journal

T. X. Li et al.The organization of this paper is as follows: Firstly, notations and preliminaries are introduced in Section 2, which are useful throughout this paper. InSection 3, we give the design of controllers and stability analysis of a multi-agentsystem is carried out. Then in Section 4, a simulation is presented to demonstrate the effectiveness of proposed technique. Finally, the conclusion is made.2. Systems DefinitionConsider the multi-agent linear system:x i ( t ) Axi ( t ) Bui ( t ) di , i 1, , N(1)where xi ( t ) R n , ui R n are the state and control input of the “ith” agent,di R n 1 is disturbance term, A and B R n n are constant matrices and the ini-tial state is defined by xi ( 0 ) . The control targets are xi xiΥ , for i 1, , N .xiY is an ideal instruction.Assumption 1. This study deals with the information exchange among agentsis modeled by an undirected graph. We assume that the communication topology is connected.3. The Design of Controllersxi ( t ) xiΥ ( t ) , thenWe define the tracking error as z i (t )z i ( t ) x i ( t ) x iΥ ( t ) Axi ( t ) Bui ( t ) di x iΥ ( t ) .(2)Design the tracking error as a sliding mode function, we give the design of control law asui ( t ) Fi ( t ) xi ( t ) uiΥ ( t ) uis ( t )(3)where Fi ( t ) is a state feedback gain matrix which can be obtained by designingLMI. Take forward-feedback control itemuiΥ ( t ) Fi ( t ) xiΥ ( t ) B 1 ( t ) Ai ( t ) xiΥ ( t ) B 1 ( t ) x iΥ ( t ) ,(4)sliding mode robust term uis B 1 ηi sgn ( zi ) , ηi R n 1 , di j ηi j 0 orTdi j ηi j 0 , and ηi sgn ( zi ) ηi1 sgnzi1 , ,ηin sgnzin .Hence,ui ( t ) Fi ( t ) xi ( t ) Fi ( t ) xiΥ ( t ) B 1 AxiΥ ( t ) B 1 x iΥ ( t ) B 1 ηi sgn ( zi ) Fi ( t ) zi ( t ) B 1 AxiΥ ( t ) B 1 x iΥ ( t ) B 1 ηi sgn ( zi ) (z i ( t ) Axi ( t ) B Fi ( t ) zi ( t ) B 1 AxiΥ ( t ) B 1 x iΥ ( t ) B 1 ηi sgn ( zi ) Υi di x (5))(t ) Axi ( t ) BFi ( t ) zi ( t ) AxiΥ ( t ) x iΥ ( t ) ηi sgn ( zi ) di x iΥ ( t )(6) Azi ( t ) BFi ( t ) zi ( t ) ηi sgn ( zi ) di .The following theorem holds.Theorem 1. Assume that AT Pi M iT Pi A M i 0 is true for any 1i 1, , N , where Fi ( Pi B ) M i . Then the closed-loop system consisting ofDOI: 10.4236/oalib.11083423Open Access Library Journal

T. X. Li et al.(1) and (3) is asymptotic stability.PROOF. Choose Lyapunov function Vi ziT Pi zi , where Pi diag { pij } is adiagonal matrix and pij 0 .Hence, V i( z P )′ z zTi iiP z Ti i i ( Azi BFi zi ηi sgn ( zi ) di ) Pi zi ziT Pi ( Azi BFi zi ηi sgn ( zi ) di )T ziT AT Pi zi ziT Fi T B T Pi zi ( η sgn ( zi ) di ) Pzi ziT Pi AziT(7) Z P BFi Z i z P ( ηi sgn ( zi ) di )Ti iTi i)( ziT AT Pi Fi T B T Pi Pi A Pi BFi zi ziT Ωi zi AT Pi Fi T B T Pi Pi A Pi BFi ,where Ωiz P ( ηi sgn ( zi ) di ) 0 .( η sgn ( z ) d )iiTPzi 0 andTi iRemark 1. In order to guarantee that V i 0 , Ωi 0 is a prerequisite. There-fore,AT Pi Fi T B T Pi Pi A Pi BFi 0.(8)In LMI (8), Fi and Pi are both uncertain. We make linearization of (3), let,M i Pi BFi then LMI (8) becomes asAT Pi M iT Pi A M i 0.(9)Making use of LMI, we can get M i and Pi , thus. Fi ( Pi B ) M i . 14. SimulationWe focus on the multi-agent linear system in this paper, without loss of generality, we assume that the system has three agents and B is the unit matrix.According to (1), 1 0 0 B 0 1 0 , A 0 0 1 2.4 9.2 0 1 1 1 , 0 16.1 3 (10)the ideal matrix is sin ( t ) cos ( t ) sin ( t ) , the interference matrix is 50sin ( t ) d1 50 cos ( t ) , d 2 50sin ( t ) 40sin ( t ) 40 cos ( t ) , d3 40sin ( t ) 50sin ( t ) 50 cos ( t ) , 50sin ( t ) (11)corresponding to the ideal matrix. Solving LMI (9), letDOI: 10.4236/oalib.110834200 10000 P1 0100000 , 0010000 (12)00 100000 P2 01000000 , 00100000 (13)4Open Access Library Journal

T. X. Li et al.00 5000000 ,P3 050000000 005000000 (14) 5.10000 37418.0101 ,F1 5.1000 37419.41017.5500 07.5500 37423.4101 (15) 5.10000 3739.6906 F2 3741.09067.5500 , 5.1000 07.5500 3745.0906 (16)0 72.4035 5.1000 F3 5.1000 73.8035 7.5500 ,07.5500 77.8035 (17)we can obtain thatFigure 1. State tracking and control input for agent 1.Figure 2. State tracking and control input for agent 2.DOI: 10.4236/oalib.11083425Open Access Library Journal

T. X. Li et al.Figure 3. State tracking and control input for agent 3.respectively. Due to (3), let 50 η1 50 , η2 50 40 40 , η3 40 50 50 , 50 (18)replacing switching function with saturation function and choosing the boundary layer as 0.05 . We give simulations are in the following (Figures 1-3).It is clear that from three figures the closed-loop system with disturbance isasymptotic stability, hence, the proposed method is effective.5. ConclusionThe multi-agent linear system was studied in this paper. Based on linear matrixinequality technology and sliding mode control, the forward-feedback controlterm was given. Sufficient conditions for the closed-loop system were establishedby Lyapunov stability theory. Simulations show that the proposed method waseffective.AcknowledgementsThe authors would like to thank the associate editor and the reviewers for theirconstructive comments and suggestions which improved the quality of the paper.Conflicts of InterestThe authors declare no conflicts of interest.ReferencesDOI: 10.4236/oalib.1108342[1]Liu, J.K. (2017) Sliding Mode Control and MATLAB Simulating. The Basic Theoryand Design Method. 3rd Edition, Peking University Press, Beijing.[2]Liu, J.K. (2017) Robot Control System Design and Matlab Simulation the Advanced6Open Access Library Journal