張佳玲，包俊東

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Simultaneouscontrol for a collection of multi-input nonlinear systems with time-delay and uncertainty

ZHANG Jia-ling，BAO Jun-dong

（School of Mathematics Science，Inner Mongolia Normal University，Huhhot 011517，China）

Researched on the problem of designing astate feedback controller that simultaneously stabilizes a collection of multi-input nonlinear systems with time-delay and uncertainty．Based on the technique of control Lyapunov-Krasovskii functional，a sufficient condition for the existence of simultaneouscontroller is obtained．A formula for constructing a continuous controller which simultaneously stabilizes the systems is derived．Proposes a method of constructing a control Lyapunov-Krasovskii functional．An example is given to show the effectiveness of the method．

multi-input nonlinear time-delay systems；simultaneous stabilization；control Lyapunov-Krasovskii functional；state feedback

1 Introduction

In recent years，simultaneouscontrol problems without time-delay aroused great attention．For linear systems，there are much achievement[1-4]．Based on a dynamic programming method，reference [1] presented a necessary and sufficient condition for the simultaneouscontrol via nonlinear digital output feedback．Reference [2] showed that，for tow linear systems，simultaneouscontrol problem is equivalent to a strongcontrol problem．For nonlinear systems，reference[5] firstly defined the new concept of control storage function（CSF）and based on the CSF method，a continuous simultaneous feedback law was developed．Based on reference [5-6] dealt with the simultaneouscontrol problem，proposed another method of constructing common quadratic control storage function，designed a continuous state feedback controller．In reference [7]，a new condition for nonlinear term was put forwarded，and under the new condition，the solvable condition of delay-dependentcontrol problem is given by using the linear matrix inequality approach．

2 Problem formulation

In the sequel，we introduce the concept of control Lyapunov-Krasovskii functional （CL-KF） for system（1）.

Definition 1A differentiable，radially unbounded，and positive definite functionalis a control Lyapunov-Krasovskii functional（CL-KF）for，if for any，with given，for，have

In next section，we proposes a method of constructing a control Lyapunov-Krasovskii functional for．

3Construction of CL-KF

Firstly，we choose a candidate Lyapunov-Krasovskii functional as

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4The construction of common feedback controller

In this part of our paper，a sufficient condition for the existence of simultaneouscontroller will be derived by the CL-KF approach．In addition，a universal，simultaneouscontroller will be developed．

For convenience to present the main results，we define some auxiliary functions firstly．For each，

Theorem 2 If there is a positive number，when

then，there exists a common state feedback controllersuch that internally stabilizes the set of systems in expression （1）simultaneously；and moreover，with given，the resultant closed-loop systems，starting from the initial state，satisfy．for all，and for each

ProofDefine

Let′s formulate the feedback controller as following

In the final，we show thatis the common controller such that internally stabilizes the set of systems in xpression

（1）simultaneously；and moreover，with given，the resultant closed-loop systems，starting from the initial state，satisfy，for all，and for each，

Firstly，we show that for any，the inequalityholds．

Tow case are considered．

from expression（10），we also have，where

or

So expression（13）is

Next，let′s proof that the resultant closed-loop systems are stabilized simultaneously．

5Example

Consider the following three multi-input nonlinear time-delay systems

6Conclusion

In this note，a novel approach has been introduced for deriving a common simultaneousstate feedback controller for a collection of multi-input nonlinear systems with time-delay and uncertainty．The authors firstly defined the new concept of CL-KF，introduced an approach for constructing the CL-KF for time-delay systems．Based on the definition of CL-KF，a sufficient condition for the existence of simultaneouscontroller is obtained，and a universal formula for constructing a continuous delay-dependent controller which simultaneously stabilizes the systems is derived．

Reference：

[1] Savkin A V．Simultaneous control of a finite collection of liner plants with a single nonlinear digital controller[J]．Syst Control Lett， 1998，33：281-289

[2] Cao Y Y，Lam J．On simultaneouscontrol and strongstabilization[J]．Automatica，2000，36：859-865

[3] Lee P H，Soh Y C．Simultaneousstabilization[J]．Int J Control，2004，77：111-117

[5] Wu Jenq-lang．Simultaneouscontrol for nonlinear systems[J]．IEEE Transactions Automatic Control，2009，54（3）：606-610

[6] Cai Xiu Shan，Gao Hong，Liu Yang．Simultaneousstabilization for a class of multi-input nonlinear systems[J]．Acta Automatica Sinica，2012，38（3）：473-478

[7] Xin Yun-bing．Delay dependentcontrol for a class of nonlinear delay systems[J]．Mathematics in Practice and Theory，2008， 38（18）：201-206

[8] Guan Xin-ping，Luo Xiao-yuan，Duan Guang-ren．Robustcontrol of time-delay uncertain systems—method of augmented Riccati equation[J]．Journal of Systems Engineering and Electronics，2002（3）：68-72

[9] Li Zhi-hu，Wang Jing-cheng，Shao Hui-he．robust control of discrete time-delay systems with time-varying uncertain[J]．Control Theory and Application，2003，20（1）：139-142

多輸入非線性不確定時滯系統(tǒng)的同時控制

張佳玲，包俊東

（內(nèi)蒙古師范大學(xué) 數(shù)學(xué)科學(xué)學(xué)院，內(nèi)蒙古 呼和浩特 011517）

所考慮的問題是設(shè)計一個狀態(tài)反饋控制器來同時鎮(zhèn)定一族多輸入非線性不確定時滯系統(tǒng)．利用控制泛函（Control Lyapunov-Krasovskii Functional）的方法給出了同時控制器存在的一個充分條件，并且給出了控制泛函及同時控制器的構(gòu)造方法，數(shù)值仿真驗證了所提方法的有效性．

多輸入非線性時滯系統(tǒng)；同時鎮(zhèn)定；控制泛函；狀態(tài)反饋

O231.1Document code：A

doi：10.3969/j.issn.1007-9831.2016.01.003

Received date：2015-10-10

Foundation item：Inner Mongolia Normal University postgraduate innovation foundation item（CXJJS14054）；Inner Mongolia Normal University ten hundred thousand talents subsidize project

Biography：ZHANG Jia-ling（1990-），female，native of Chifeng of Inner Mongolia，postgraduate，main research field：systems with time delay stabilization and robust control．E-mail：zybjl818@163.com