Performance Analysis,Distributed Control And Filtering For Spatially Interconnected Systems

Spatially interconnected systems consist of similar units which interact with their nearest neighbors in such a way that the input and output signals depend not only on time but also on some spatial dimensions.The resulting system brings a lot of restrictions in control theory.Hence,in recent years,spatially interconnected systems have been investigated by many scholars,since this kind of system has become a significant research theme due to the complicated properties in subsystems and wide applications in aerial vehicles formations,airplane formation flight,vehicles platoon in automated highways systems,cross-directional control in sheet and film processing,lumped approximations of partial differential equations(PDEs),and so on.Based on stability theory,Lyapunov function,operator theory,distributed control and other design tools,the distributed control design of spatially interconnected systems has achieved remarkable results.This paper will further design and analyze the distributed filtering,the distributed stochastic control and the distributed finite-time control for spatially interconnected systems with uncertainties and Markovian jumping parameters.All results are based on linear matrix inequalities(LMIs).The main contents include:(1)Robust distributed filtering problem is investigated for spatially interconnected systems with parametric uncertainties.First,a sufficient condition based on the partitioned system matrices is given to guarantee the well-posedness,quadratic stability and contractiveness of the spatially interconnected systems with LFT uncertainties.Due to the property of this kind of systems,a modified bilinear transformation is used to convert a discrete time and discrete space system to an equivalent continuous time and continuous space system.Thus,an equivalent sufficient condition which is directly applied to the design of the robust distributed filters is shown without the partition of the system matrices and the desired robust distributed filters are obtained.In the end,a vehicle platoon system example demonstrates the effectiveness of the proposed scheme.(2)The distributed stochastic control problem is concerned for spatially interconnected Markovian jump systems with time-varying delays.Similarly,a modified bilinear transformation is introduced to convert a continuous time and discrete space system to an equivalent continuous time and continuous space system.Combined with the stochastic Lyapunov function,the existence condition of the expected distributed stochastic controllers is presented.In the end,an illustrative numerical example is given to demonstrate the effectiveness of the results.(3)The distributed finite-time control problem is addressed for spatially interconnected systems.Compared with the classical Lyapunov asymptotic stability,finite-time stability is much more important and significant in some practical applications.First,the definitions of finite-time stability/boundedness and finite-time contractiveness for spatially interconnected systems are introduced for the first time.Second,a sufficient condition is presented to check the well-posedness,finite-time stability/bounedness and finite-time contractiveness of spatially interconnected systems based on the given system matrices,which plays an important part in the distributed finite-time control scheme.Third,the existence condition of the distributed controllers is proposed by making use of the given system matrices,which is easy to be implemented.Besides,a one-dimensional heat equation example is demonstrated to verify the effectiveness of the proposed control method.(4)The distributed finite-time stochastic control problem is studied for spatially interconnected Markovian systems.First of all,the concepts of well-posedness,finitetime stochastic stability/boundedness and finite-time stochastic contractiveness are introduced.Moreover,a significant sufficient condition is explored to ensure that the expectation of system state stays in a fixed bound in a finite-time interval under a preset initial bound.Furthermore,the formulation of distributed stochastic controllers is developed such that the corresponding closed-loop spatially interconnected Markovian jump systems are well-posed,finite-time stochastic stable/bounded and finite-time stochastic contractive.In the end,a one-dimensional heat equation simulation result is employed to clarify the validity and effectiveness of the given control scheme.