Research On Optimal Stabilization Of Discrete-time Switched Linear Systems

Lots of systems with the peculiarity of interaction between the continuous dynamics and discrete events,this kind of systems are termed as Hybrid Systems.In recent decades,with the developing of applied mathematics,computer science and control science,and the rise of interdisciplinary research,a significant progress has been made in the research on the hybrid systems.As a class of important hybrid systems,a switched system consists of a finite number of subsystems and a switching rules governing the activeness sequence of those subsystems.Due to the introduction of switching rules,a switched system can not only maintain partial performance of each subsystem,but also may show some complex performance that each subsystem does not have.Therefor,switched systems can be used to describe many complex nonlinear systems.Switched linear systems is an important type of switched systems.The research on it can not only adopt the full-blown linear system theory,but also helpful for the study of more complex switched/hybrid systems.Therefore,the research on switched linear systems is a hot field.Stability/stabilizability is the all-important problem of switched linear systems.Thus,for the past decades,researchers have made a lot of researches on it,and have established a relatively perfect theories system.However,the switching law and/or the control input designed for the asymptotic performance may cause a large overshoot/or high frequency oscillation,which may damage the system.Therefore,it is an important problem to design a switching law and/or a control input with acceptable transient and asymptotic performance.This is the optimizing stabilization design problem of switched linear systems.In this dissertation,using the overshoot/peak as the optimizing index,the optimizing stabilization of discrete-time switched linear systems is studied.Specifically,the main research contents and results are as follows.1.Stabilization design of discrete-time autonomous switched linear systems with state-overshoot as optimizing index.Based on the research of finite time minimum stateovershoot problem,we modify a so-called state-feedback path-wise switching law,an optimal asynchronous stabilizing switching law is obtained,and the range of the system state is estimated.In the asynchronous approach,we construct the switching signal by the concatenation of different length of switching paths,which brings some inconvenience to the engineering implementation of the switching law.In view of this,we propose that the switching signal is constructed by the concatenation of the same length of the switching paths,which is the synchronous optimal stabilizing the switching law.In the synchronous approach,the first thing is to determine the appropriate length of each switching path.Obviously,the length of the switching paths satisfying the state norm diminish condition only exists minimum value,and does not exist maximum value.In addition,with the increase of the length of the switching paths,the calculation burden of searching a set of switching paths meeting the norm constraint condition significantly increase,however,the system state-overshoot does not necessarily reduce.Therefore,we should adopt a set of switching paths with small length and meeting the norm constraint condition at the same time.By considering two kinds of limit cases,we have proved the range of the minimum length of the switching paths in the synchronous approach,which lays the theoretical foundation for the rapid determination of the length of the appropriate switching paths.Once the length of the switching paths is determined,we can design the optimizing switching law similar to the asynchronous approach.We have proved the exponential asymptotic stability of the synchronous approach,and estimated the upper bound of the system state-overshoot.Although we have proved the range of the minimum length of switching paths in the synchronous approach,to get the appropriate switching path length,still need to go through some attempts,this is not convenient.According to the switched convergence of a matrix set,and using the integer programming,we set up a rapid approach for the determination of minimum length of switching paths for switched systems with diagonal or triangular subsystem matrices.2.Stabilization design of discrete-time autonomous switched linear systems with output-peak as optimizing index.As the output at time 0 is not only dependent on the initial state,but also connected with the activated subsystem at this time.Therefore,the output-overshoot optimization problem is different from the output-peak one,the former is very complicated,we study it later,and only address the output-peak optimizing stabilization problem.We address this problem by using the train of thought similar to that used to address the state-overshoot optimizing stabilization one.Specifically,we first study some properties of the finite time minimum output-peak problem,and then design the asynchronous and synchronous output-peak optimizing stabilization switching law,finally,the range of the output-peak is estimated.In addition,we consider the output convergence from the point of view of the switched convergence of a matrix set pair,which lays a theoretical foundation for further research on output optimizing stabilization.3.State/output non-overshoot stabilization problem of discrete-time forced switched linear systems.For this problem,as both the switching law and control input are the variables to be designed,generally speaking,the design of these two variables coupling with each other,which makes the problem quite complicated.However,we found that when the ?2-norm is used to measure overshoot,the non-overshoot stabilization problem can be addressed handily.Specifically,based on the solution of the quadratic minimal problem,we first study the non-overshoot condition of a subsystem.In general,each subsystem can not non-overshoot in the whole state space,but only in a pair of 0-symmetric cones centered on the origin of the state space,which are defined as the state non-overshoot region of the subsystem.It is not hard to think that if the union of each state non-overshoot region of a switched system covering the whole state space,then under an appropriate switching law and a control input,the state non-overshoot can be realised at arbitrary initial condition.The problem is how to verify the full coverage.In view of the mathematical expression of a non-overshoot region,we adopt the S-procedure to deal with the problem,and the full coverage problem is transformed into a feasibility problem of a linear matrix inequality,which is easy to verify.Moreover,when using this approach,the design of the switching law and the design of the control input can be carried out almost independently with each other,which is what we expect.As for the output non-overshoot problem,we first transform it into a state space problem,and then use the approach which is similar to that used to address the state non-overshoot one,and similar results are obtained.The difference is that,to make the monotonic decrease of the output ?2-norm,a guaranteed condition should be added.The main theoretical results are verified by simulation examples.Finally,the conclusions and the problems to be explored in the future are given at the end of this dissertation.