H∞ and BIBO stabilization of delay systems of neutral type

Adaptive control of dynamic mobile robots withnonholonomic constraintsFarzad Pourboghrat *,Mattias P.KarlssonDepartment of Electrical and Computer Engineering,Southern Illinois University,Carbondale,IL 62901-6603,USAReceived 16November 1999;accepted 22August 2000AbstractThis paper presents adaptive control rules,at the dynamics level,for the nonholonomic mobile robots with unknown dynamic parameters.Adaptive controls are derived for mobile robots,using backstepping technique,for tracking of a reference trajectory and stabilization to a fixed posture.For the tracking problem,the controller guarantees the asymptotic convergence of the tracking error to zero.For stabili-zation,the problem is converted to an equivalent tracking problem,using a time varying error feedback,before the tracking control is applied.The designed controller ensures the asymptotic zeroing of the sta-bilization error.The proposed control laws include a velocity/acceleration limiter that prevents the robot Õs wheels from slipping.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Mobile robot;Nonholonomic constraint;Dynamics level motion control;Stabilization and tracking;Adaptive control;Backstepping technique;Asymptotic stability1.IntroductionMotion control of mobile robots has found considerable attention over the past few years.Most of these reports have focused on the steering or trajectory generation problem at the ki-nematics level i.e.,considering the system velocities as control inputs and ignoring the mechanical system dynamics [1–3].Very few reports have been published on control design in the presence of uncertainties in the dynamic model [4].Some preliminary results on control of nonholonomic systems with uncertainties are given in Refs.[4–6].Two of the most important control problems concerning mobile robots are tracking of a refer-ence trajectory and stabilization to a fixed posture.The tracking problem has received solutions *Corresponding author.Tel.:+1-618-453-7026.E-mail address:pour@ (F.Pourboghrat).0045-7906/02/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S0045-7906(00)00053-7242 F.Pourboghrat,M.P.Karlsson/Computers and Electrical Engineering28(2002)241–253including classical nonlinear control techniques[1,2,7].The basic idea is to have a reference car that generates a trajectory for the mobile robot to follow.In Refs.[1,2],nonlinear velocity control inputs were defined that made the tracking error go to zero as long as the reference car was moving.In Ref.[7],they used input–output linearization to make a mobile platform follow a desired trajec-tory.The problem of stabilization about afixed posture has been shown to be rather complicated. This is due to violating the BrockettÕs condition[8],which states that for nonholonomic systems a single equilibrium solution cannot be asymptotically stabilized using continuous static state feedback[9,10].The BrockettÕs condition essentially states that for nonholonomic systems an equilibrium solution can be asymptotically stabilized only by either a time varying,a discontin-uous,or a dynamic state feedback.In addressing the above problem,in Ref.[10]a smooth feedback control was presented for the kinematics control problem resulting in a globally marginally stable closed loop system.They also designed a smooth feedback control for a dynamical state-space model resulting in a Lagrange stable closed loop system,as defined in their paper.A two dimensional Lyapunov function was utilized in Ref.[3]to prescribe a set of desired trajectories to navigate a mobile robot to a specified configuration.The desired trajectory was then tracked using sliding mode control,resulting in discontinuous control signals.The mobile robot was shown to be exponentially stable for a class of quadratic Lyapunov functions.In Ref.[9],they formulated a reduced order state equation for a class of nonholonomic systems.Several other researchers have later used this reduced order state equation in their studies.In Ref.[4],the problem of controlling nonholonomic mechanical sys-tems with uncertainties,at the dynamics level,was ing the reduced state equation in Ref.[9],they proposed an adaptive controller for a number of important nonholonomic control problems,including stabilization of general systems to an equilibrium manifold and stabilization of differentiallyflat and Caplygin systems to an equilibrium point.In Ref.[2],they gave several examples on how the stabilization problem can be solved for a mobile robot at the kinematics level.Their solutions included time-varying control,piecewise continuous control,and time-varying piecewise continuous control.They also showed how a solution to the tracking problem could be extended to work even for the stabilization problem.Here,we present adaptive control schemes for the tracking problem and for the problem of stabilization to afixed posture when the dynamic model of the mobile robot contains unknown parameters.Our work is based on,and can be seen as an extension of,the work presented in Refs. [1,2].Using backstepping technique we derive adaptive control laws that work even when the model of the dynamical system contains uncertainties in the form of unknown constants.The assumption for the uncertainty in robotÕs parameters,particularly the mass,and hence the inertia, can be justified in real applications such as in automotive manufacturing industry and warehouses, where the robots are to move a variety of parts with different shapes and masses.In these cases,the robotÕs mass and inertia may vary up to10%or20%,justifying an adaptive control approach.2.Dynamic model of mobile robotHere,we consider a three-wheeled mobile robot moving on a horizontal plane(Fig.1).The mobile robot features two differentially driven rear wheels and a castor front wheel.The radius ofthe wheels is denoted r and the length of the rear wheel axis is 2l .Inputs to the system are two torques T 1and T 2,provided by two motors attached to the rear wheels.The dynamic model for the above wheeled-mobile robot is given by Refs.[10,11].€x ¼k m sin /þb 1u 1cos /€y ¼Àk cos /þb 1u 1sin /€/¼b 2u 28><>:ð1Þ_x sin /À_y cos /¼0ð2Þwhere b 1¼1=ðrm Þ,b 2¼l =ðrI Þ,and that m and I denote the mass and the moment of inertia of the mobile robot,respectively.Also,u 1¼T 1þT 2and u 2¼T 1ÀT 2are the control inputs,and k is the Lagrange multiplier,given by k ¼Àm _/_x cos /þ_y sin /ðÞ.Here,it is assumed that b 1and b 2are unknown constants with known signs.The assumption that the signs of b 1and b 2are known is practical since b 1and b 2represent combinations of the robot Õs mass,moment of inertia,wheel radius,and distance between the rear wheels.Eq.(2)is the nonholonomic constraint,coming from the assumption that the wheels do not slip.The triplet vector function q t ðÞ¼x t ðÞ;y t ðÞ;/t ðÞ½ T denotes the trajectory (position and orientation)of the robot with respect to a fixed workspace frame.That is,at any given time,q ¼½x ;y ;/ T describes the robot Õs configuration (posture)at that time.We assume that,at any time,the robot Õs posture,q ¼½x ;y ;/ T ,as well as its derivative,_q¼½_x ;_y ;_/ T ,are available for feedback.3.Tracking problem definitionThe tracking problem consists of making the trajectory q of the mobile robot follow a reference trajectory q r .The reference trajectory q r t ðÞ¼x r t ðÞ;y r t ðÞ;/r t ðÞ½ T is generated by a reference ve-hicle/robot whose equations are_xr ¼v r cos /r _y r¼v r sin /_/r ¼x r8<:ð3ÞThe subscript ‘‘r’’stands for reference,and v r and x r are the reference translational (linear)velocity and the reference rotational (angular)velocity,respectively.We assume that v r and x r ,as well as their derivatives are available and that they all are bounded.Assumption A 1.For the tracking problem it is assumed that the reference velocities v r and x r do not both go to zero simultaneously.That is,it is assumed that at any time either lim t !1v r t ðÞ90and/or lim t !1x r t ðÞ90.The tracking problem,under the Assumption A 1,is to find a feedback control law u 1u 2 ¼u q ;_q ;q r ;v r ;x r ;_v r ;_x r ðÞsuch that lim t !1~q t ðÞ¼0,where ~q t ðÞ¼q r t ðÞÀq t ðÞis defined as the trajectory tracking error.As in Ref.[1],we define the equivalent trajectory tracking error ase ¼T ~qð4Þwhere e ¼½e 1;e 2;e 3 T ,and T ¼cos /sin /0Àsin /cos /00010@1A .Note that since T matrix is nonsingular,e is nonzero as long as ~q¼0.Assuming that the angles /r and /are given in the range ½Àp ;p ,we have the equivalent trajectory tracking error e ¼0only if q ¼q r .The purpose of the tracking controller is to force the equivalent trajectory tracking error e to 0.In the sequel we refer to e as the trajectory tracking error.Using the nonholonomic constraint (2),the derivative of the trajectory tracking error given in Eq.(4)can be written as,[1],_e1¼e 2x Àv þv r cos e 3_e 2¼Àe 1x þv r sin e 3_e3¼x r Àx 8<:ð5Þwhere v and x are the translational and rotational velocities of the mobile robot,respectively,and are expressed asv ¼_xcos /þ_y sin /x ¼_/ð6Þ4.Tracking controller designHere,the goal is to design a controller to force the tracking error e ¼½e 1;e 2;e 3 T to ing backstepping technique,since the actual control variables u 1and u 2do not appear in Eq.(5),we consider variables v and x as virtual controls.Let v d and x d denote the desired virtual controls for the mobile robot.That is,with v d and x d the trajectory tracking error e converges tozero asymptotically.Also let us define ~vand ~x as virtual control errors.Then,v and x can be written asv ¼v d þ~vx ¼x d þ~x ð7Þ244 F.Pourboghrat,M.P.Karlsson /Computers and Electrical Engineering 28(2002)241–253Let us choose the virtual controls v d and x d ,asv d v r ;x r ;e 1;e 3ðÞ¼v r cos e 3þk 1v r ;x r ðÞe 1x d v r ;x r ;e 2;e 3ðÞ¼x r þk 2v r e 2þk 3v r ;x r ðÞsin e 3ð8Þwhere k 2is a positive constant and k 1ðÁÞand k 3ðÁÞare bounded continuous functions with bounded first derivatives,strictly positive on R ÂR -ð0;0Þ.Observe that our approach from here on is general for any v d and x d (with well defined first derivatives),i.e.any differentiable control law that makes the kinematics model of the mobile robot track a desired trajectory can be used instead of Eq.(8).Eq.(8)is similar to the control law proposed by Ref.[1],but with the advantage,as we are going to prove later,that it can be used to track any reference trajectory as long as As-sumption A 1holds.Now,consider the following adaptive control scheme:u 1¼^b 1ðÀc 1~v þe 1þ_v d Þu 2¼^b 2 Àc 2~x þ1k 2sin e 3þ_x d _^b 1¼Àc 1sign b 1ðÞ~v ðÀc 1~v þe 1þ_v d Þ_^b 2¼Àc 2sign b 2ðÞ~x Àc 2~x þ1k 2sin e 3þ_x d ð9Þwhere c 1,c 2,c 1,and c 2are positive constants and ^b 1is an estimate of b 1¼1=b 1and ^b 2is an estimate of b 2¼1=b 2.Result 1.If Assumption A 1holds ,then the adaptive control scheme (9)makes the origin e ¼0uniformly asymptotically stable.Proof .Consider the following Lyapunov function candidateV 1¼12e 21Àþe 22Áþ1k 21ðÀcos e 3Þð10Þwhere k 2is a positive constant.Clearly V 1is positive definite and V 1¼0only if e ¼0.Taking the time derivative of V 1,we obtain_V 1¼e 1ðÀv þv r cos e 3Þþe 2v r sin e 3þ1k 2sin e 3x r ðÀx Þð11ÞFurthermore,using Eqs.(7)and (8),we have_V 1¼Àk 1e 21Àk 3k 2sin 2e 3À~v e 1À~x 1k 2sin e 3ð12ÞIn view of Eqs.(1),(2)and (6),we find the time derivatives of ~vand ~x ,as _~v¼_v À_v d ¼€x cos /À_x sin /_/þ€y sin /þ_y cos /_/À_v d ¼b 1u 1À_v d _~x ¼_x À_x d ¼€/À_x d ¼b 2u 2À_x d ð13ÞF.Pourboghrat,M.P.Karlsson /Computers and Electrical Engineering 28(2002)241–253245Consider the Lyapunov function candidateV2¼V1þ12ð~v2þ~x2Þþb1j j2c1~b21þb2j j2c2~b22ð14Þwhere~b1¼b1À^b1¼1=b1À^b1and~b2¼b2À^b2¼1=b2À^b2.Considering Eq.(9)we get:_V 2¼Àk1e21Àk3k2sin2e3Àc1~v2Àc2~x260ð15ÞSince V2is bounded from below and_V2is negative semi-definite,V2converges to afinite limit. Also,V2,as well as,e1,e2,e3,~v,~x,^b1,and^b2are all bounded.Furthermore,using Eqs.(5),(7)–(9)and(13),the second derivative of V2can be written as€V 2¼À2k1e1e2ðx rþk2v r e2þk3sin e3þ~xÞþ2k1e1ðk1e1þ~vÞÀ_k1e21þ2k3k2cos e3sin e3ðk2v r e2þk3sin e3þ~xÞÀ_k3k2sin2e3À2c1~vðb1^b1ðÀc1~vþe1þ_v dÞÀ_v dÞÀ2c2~x b2^b2Àc2~xþ1k2sin e3þ_x dÀ_x dð16Þwhich from the properties of k1,k2,and k3,the assumption that v r and x r and their derivatives are bounded,and from the above results,can be shown to be bounded,i.e.,_V2is uniformly contin-uous.Since V2ðtÞis differentiable and converges to some constant value and that€V2is bounded,by BarbalatÕs lemma,_V2tðÞ!0as t!1.This in turn implies that e1,e3,~v,and~x converge to zero [12,13].To show that e2also goes to zero,note that,using the above results,thefirst error equation can be written as_e1¼e2x rÀk1e1ð17ÞThe second derivative of e1is€e1¼_x r e2þx rðÀe1xþv r sin e3ÞÀk1e2x rðÀk1e1ÞÀ_k1e1ð18Þwhich can be shown to be bounded by once again using the properties of k1,the assumptions on v r and x r,and Eqs.(7)and(8).Since e1is differentiable and converges to zero and€e1is bounded,by BarbalatÕs lemma,_e1,and hence,e2x r tend to zero.Proceeding in the same manner,the third error equation can be written as_e3¼Àk2v r e2Àk3sin e3ð19Þand its second derivative can be shown to be bounded.Since e3is differentiable and converges to zero and€e3is bounded,again by BarbalatÕs lemma,_e3!0as t!1.Hence,k2v r e2and thus v r e2 tend to zero as t!1.Clearly,both v r e2and x r e2converge to zero.However,since v r and x r do not both tend to zero(by Assumption A1),e2must converge to zero.That is,e1,e2,e3,~v,and~x must all converge to zero.hIn Section3,we demonstrated that the system is stable if k2is a positive constant,and that k1ðÁÞand k3ðÁÞare bounded continuous functions with boundedfirst derivatives and are strictly positive on RÂR-ð0;0Þ.To get a better understanding on how the control gains affect the response of the system,we write the equations for the closed loop system when~v and~x are equal to zero as[1] 246 F.Pourboghrat,M.P.Karlsson/Computers and Electrical Engineering28(2002)241–253_e¼Àk 1e 1þx r þk 2v r e 2þk 3sin e 3ðÞe 2Àx r þk 2v r e 2þk 3sin e 3ðÞe 1þv r sin e 3Àk 2v r e 2Àk 3sin e 30@1A ð20ÞBy linearizing the differential equation (20)around e ¼0,we get_e¼Ae ð21Þwhere A ¼Àk 1x r 0Àx r 0v r 0Àk 2v r Àk 30@1A ð22ÞTo simplify the analysis,we assume that v r and x r are constants.The system Õs closed loop poles are now equal to the roots of the following characteristic polynomial equation:s ðþ2nx 0Þs 2Àþ2nx 0s þx 20Áð23Þwhere n and x 0are positive real numbers.The corresponding control gains arek 1¼2nx 0k 2¼x 20Àx 2r v 2r k 3¼2nx 0ð24ÞWith a fixed pole placement strategy (n and x 0are constant),the control gain k 2increaseswithout bound when v r tends to zero.One way to avoid this is by letting the closed loop polesdepend on the values of v r and x r .As in Ref.[2],we choose x 0¼x 2r þbv 2r ÀÁð1=2Þwith b >0.The control gains then becomek 1¼2n x 2r Àþbv 2rÁ1=2k 2¼b k 3¼2n x 2r Àþbv 2r Á1=2ð25Þand the resulting control is now defined for any values of v r and x r .In the above,it is shown that the proposed algorithm works for any desired velocities,ðv d ;x d Þ.However,in practice,if the tracking errors initially are large or if the reference trajectory does not have a continuous curvature (e.g.,if the reference trajectory is a straight line connected to a circle segment),either or both of the virtual reference velocities in Eq.(8)might become too large for a real robot to attain in practice.Hence,the translational/rotational acceleration might become too large causing the robot to slip [1].In order to prevent the mobile robot from slipping,in a real application,a simple velocity/acceleration limiter may be implemented [1],as shown in Fig.2.This limits the virtual reference velocities ðv d ;x d Þby constants ðv max ;x max Þand the virtual referenceaccelerations ða ;a Þby constants ða max ;a max Þ,where a ¼_vd and a ¼_x d are the virtual reference accelerations.In practice,these parameters must be determined experimentally as the largest values with which the mobile robot never slips.F.Pourboghrat,M.P.Karlsson /Computers and Electrical Engineering 28(2002)241–253247An important advantage of adding the limiter is that it lowers the control gains indirectly only when the tracking errors are large,i.e.,when too high a gain could cause the robot to slip,while for small tracking errors it does not affect the performance at all.Thus,by using the limiter one can have higher control gains for small tracking errors to allow for better tracking,while letting the limiter to‘‘scale down’’the gains,indirectly,for large tracking errors,to prevent the robot from slipping.5.Simulation results for tracking control problemHere,the results of computer simulation,using MATLAB/SIMULINK,are presented for a mobile robot with the proposed tracking control and with the velocity/acceleration limiter.The computer simulations for the above controller without the limiter,although not shown here,produce similar results,but with somewhat different transient characteristics.All simulations have the common parameters of c1¼c2¼100,c1¼c2¼10and b¼250.Also selected are,the damping factor n¼1,v max¼1:5m/s,x max¼3rad/s,a max¼5m/s2and a max¼25rad/s2.Moreover,the robotÕs dy-namic parameters are chosen as b1¼b2¼0:5,which are assumed to be unknown to the con-troller,but with known signs.Simulation results for the case where the reference trajectory is a straight line are shown in Figs. 3and4for t2½0;10 .The reference trajectory is given by x rðtÞ¼0:5t,y rðtÞ¼0:5t and/rðtÞ¼p=4, defining a straight line,starting from q rð0Þ¼½x rð0Þ;y rð0Þ;/rð0Þ T¼½0;0;p=4 T.The mobile robot, however,is initially at qð0Þ¼½xð0Þ;yð0Þ;/ð0Þ T¼½1;0;0 T,where/¼0indicates that the robot is heading toward positive direction of x.As it can be seen from thesefigures,first the robot backs up and then heads toward the virtual reference robot moving on the straight line.Figs.5and6show the simulation results for tracking a circular trajectory.The reference trajectory is a point moving counter clockwise on a circle of radius1,starting at q rð0Þ¼½x rð0Þ;y rð0Þ;/rð0Þ T¼½1;0;p=2 T.The reference velocity is kept constant at v rðtÞ¼0:5m/s.The initial conditions for the mobile robot,however,is taken as qð0Þ¼½xð0Þ;yð0Þ;/ð0Þ T¼½0;0;0 T.Again,as it is seen from thesefigures,the robot immediately heads toward the reference robot,which is moving on the circle.It then reaches it quickly and continues to track it.6.Stabilization problem definitionThe stabilization problem,given an arbitrary desired posture q d ,is to find a feedback control law,u 1u 2¼u q Àq d ;_q ;t ðÞ,such that lim t !1q t ðÞÀq d ðÞ¼0,for any arbitrary initial robot posture q ð0Þ.Without loss of generality,we may take q d ¼½0;0;0 T.6.1.Stabilization controller designRecall that there is no continuous static state feedback that can asymptotically stabilize a nonholonomic system about a fixed posture [8–10].The approach to the problem taken here is the dynamic extension of that in Ref.[2]where a kinematics model of the mobile robot is used.In-stead of designing a new controller for the stabilization problem the same controller as for the tracking problem is used.The idea is to let the reference vehicle move along a path that passes through the point ðx d ;y d Þwith heading angle /d .The stabilization to a fixed posture problem isnow equivalent to,and can be treated as,a tracking problem(convergence of the tracking errors to zero)with the additional requirement that the reference vehicle should itself be asymptotically stabilized about the desired posture.As in Ref.[2],we let the reference vehicle move along the x-axis,i.e.y rðtÞ¼0and/rðtÞ¼0,for all values on t.The design method is the same as derived for the tracking case.However,in this casev r¼_x r¼Àk4x rþgðe;tÞ;ð26Þwithgðe;tÞ¼k e k sin tð27Þwhere k4>0.Different time-varying functions gðe;tÞhave also been suggested in the literature,see Refs.[2,11]and the references therein.Since,from the Section5,the tracking errors e1,e2,and e3are bounded,the time-varying function gðe;tÞis bounded.Therefore v r and the state x r also remain bounded.By taking the time derivative of Eq.(26),it can be shown in the same way that_v r is bounded.Since v r and_v r are bounded,the assumptions made in Section3concerning the reference velocity are fulfilled.If v r is not equal to zero,then e must converge to zero.When e tends to zero,gðe;tÞalso tends to zero. Therefore,the robotÕs position x must track x r,which converges to zero and hence lead the mobile robot to the desired posture.6.2.Simulation results for stabilization control problemHere,the simulation results for the stabilization problem are shown in Figs.7and8.The control parameters and system parameters are the same as for the simulations shown for the tracking problem and k4¼1.The mobile robot is initially at qð0Þ¼½xð0Þ;yð0Þ;/ð0Þ T¼½0;1;0 T.As it is seen from thefigures,the stabilization about thefinal posture at the origin is achieved quite satisfactorily.Note,in this case,that the robot actually turns around and backs up into the final posture.7.ConclusionsTwo important control problems concerning mobile robots with unknown dynamic parameters have been considered,namely,tracking of a reference trajectory and stabilization to afixed posture.An adaptive control law has been proposed for the tracking problem and has been ex-tended for the stabilization problem.A simple velocity/acceleration limiter was added to the controller,for practical applications,to avoid any slippage of the robotÕs wheels,and to improve the tracking performance.Several simulation results have been included to demonstrate the performance of the proposed adaptive control law.References[1]Kanayama Y,Kimura Y,Miyazaki F,Noguchi T.A stable tracking control method for an autonomous mobilerobot.vol.1.Proceedings of IEEE International Conference on Robotics and Automation,Cincinnati,Ohio,1990, p.384–9.[2]Canudas de Wit C,Khennouf H,Samson C,Sordalen OJ.Nonlinear control design for mobile robots.In:ZhengYF,editor.Recent trends in Mobile robots,World Scientific,1993.p.121–56.[3]Guldner J,Utkin VI.Stabilization of nonholonomic mobile robot using Lyapunov functions for navigation andsliding mode control.Control-Theory Adv Technol1994;10(4):635–47.[4]Colbaugh R,Barany E,Glass K.Adaptive Control of Nonholonomic Mechanical Systems.Proceedings of35thConference on Decision and Control,Kobe,Japan,1996.p.1428–34.[5]Fierro R,Lewis FL.Control of nonholonomic mobile robot:backstepping kinematics into dynamics.J Robot Sys1997;14(3)149-163.[6]Jiang ZP,Pomet bining backstepping and time-varying techniques for a new set of adaptive controllers.Proceedings of33rd IEEE Conf on Decision and Control,Lake Buena Vista,FL,1994.p.2207–12.[7]Sarkar N,Yun X,Kumar V.Control of mechanical systems with rolling constraints:application to dynamiccontrol of mobile robots.Int J Robot Res1994;13(1):55–69.[8]Brockett RW.Asymptotic stability and feedback stabilization.In:Brockett RW,Millman RS,Sussmann HJ,editors.Differential Geometric Control Theory,Boston,MA:Birkhauser;1983.p.181–91.[9]Bloch AM,Reyhanoglu MR,McClamroch NH.Control and stabilization of nonholonomic dynamic systems.IEEE Trans Automat Contr1992;37(11):1746–56.[10]Campion G,d’Andrea-Novel B,Bastin G.Controllability and state feedback stabilization of nonholonomicmechanical systems.Canudas de Wit C,editor.Advanced Robot Control,Berlin:Springer;1991.p.106–24. [11]Kolmanovsky I,McClamroch NH.Developments in nonholonomic control problems.IEEE Contr Sys Magaz1995;15(6):20–36.[12]Krstic M,Kanellakopoulos I,Kokotovic P.Nonlinear and Adaptive Control Design,New York:Wiley;1995.[13]Karlsson MP.Control of nonholonomic systems with applications to mobile robots.Master Thesis,SouthernIllinois University,Carbondale,IL62901,USA,1997.Farzad Pourboghrat received his Ph.D.degree in Electrical Engineering from the University of Iowa in1984. He is now with the Department of Electrical and Computer Engineering at Southern Illinois University at Carbondale(SIU-C)where he is an Associate Professor.His research interests are in adaptive and slidingcontrol with applications to DSP embedded systems,mechatronics,flexible structures andMEMS.Mattias Karlsson received the B.S.E.E.degree from the University of Bor a s,Sweden and the M.S.E.E.degree from Southern Illinois University,Carbondale,IL,in1995and1997,respectively.He has been employed at Orian Technology since1997.He is currently an on-site consultant at Caterpillar Inc.Õs Technical Center, Mossville,IL.His current interests include control algorithm development for mechanical and electrical systems and software development for embedded systems.F.Pourboghrat,M.P.Karlsson/Computers and Electrical Engineering28(2002)241–253253。

一类具有分布时滞的SIR传染病模型的稳定性分析王桂臻;李必文【摘要】研究了一类具有分布时滞的SIR传染病模型的稳定性,通过构造适当的Lyapunov泛函得到保证系统地方病平衡点全局渐近稳定的充分性条件.%The stability of an SIR epidemic model with distributed delay is studied by building appropriate Lyapunov functional,some sufficient conditions for the global asymptotically of endemic equilibrium are Obtained.【期刊名称】《湖北师范学院学报（自然科学版）》【年(卷),期】2012(032)003【总页数】4页(P50-53)【关键词】时滞;分布时滞;全局稳定性;Lyapunov泛函【作者】王桂臻;李必文【作者单位】湖北师范学院数学与统计学院,湖北黄石435002;湖北师范学院数学与统计学院,湖北黄石435002【正文语种】中文【中图分类】O6510 引言数学模型在分析传染病的传播和控制方面已经成为重要的工具.近几十年来,国际上传染病动力学的研究进展十分迅速,大量的数学模型被用于分析各种各样的传染病问题,所涉及的因素也不断增多,方程的形式越来越复杂,如引起广泛关注的时滞微分方程模型.时滞在传染病模型中是广泛存在的，Betertta等在文献中研究了一类具有时滞的SIR传染病模型,他们得到了无病平衡点的全局稳定性和地方病平衡点的局部稳定性;如果时滞充分小,地方病平衡点还是全局稳定的.这是时滞模型的一个基本结果,即地方病平衡点对于小时滞是全局稳定的.2000年,Taketlchi、Ma和Betertta又研究了另一个时滞SIR模型,得到了类似的结果.2002年,Ma和Taketlchi研究了一类具有分布时滞的SIR传染病模型,得到了只要地方病平衡点存在,模型就是持久的.在[1]中，已经研究了下面的分布时滞模型(1)其中，易感人群指定为S，染病人群为I，恢复人群R具有持久免疫力. b是人口出生率，λ 是感染人群的恢复率， ui(i=1,2,3)分别表示易感人群疫苗覆盖，β是为S 类和I类的有效接触率, f(τ)dτ=1(假定 f取非零值且任意地接近于h ).所有的参数都是正数.本文将在此基础上，用代替I(t-τ)，其中α是依赖于I的S的个体减少率的最大值，a 是半饱和常数，S,I,R 以及正常数b,λ,β,h,μi(i=1,2,3) 的生物意义完全类同系统(1)，得到我们要研究的模型(2)满足初始条件S(0)≥0,I(0)≥0,R(0)≥0.1 平衡点分析令系统(2)的右端为零, 可给出系统(2)的平衡点:定义基本再生数对任意参数b,λ,β,h,a,α，μi(i=1,2,3),我们看到系统(2)总是有一个无病平衡点如果R0≤1 ，那么E0 是唯一的平衡点；如果R0>1，也存在唯一的地方病平衡点E*(S*,I*,R*) ，其中2 全局渐近稳定性由于系统(2)的前两个方程不依赖于变量R，因此我们仅考虑前两个方程的稳定性. 定理1 若R0>1 ，那么E* 是全局渐近稳定的.证比较在E* 处系统(2)的两边以及f(τ)dτ=1,并令函数得b=βm(S*)I*+μ1S*(3)(μ2+λ)I*=βm(S*)I*(4)令g(y)=y-1-lny ，及(5)其中φ(τ)=f(σ)dσ.下面我们来研究Lyapunov泛函的表现形式，定义如下(6)注意到，当ω∈[0,h) ，有φ(ω)>0 ，并且g:R>0→R≥0 有最小值g(1)=0 .当且仅当时，W(t)=0 .下面来分别计算和以最终得到的值.用(3)替代b，得到令得到(7)下面我们来计算用(4)代替(μ2+λ)I* ，得(8)现在我们来计算用定积分的分部积分法，我们有(9)注意到，那么上式就变为f(τ)(g(y)-g(z))dτ=f(τ)(y-lny-z+lnz)dτ(10)结合式(7)，(8)，(10),可得对上式进行变形，得到容易验证lnF(x)+z-F(x)z≥0 ,且根据函数m(x) 的递增性质，可判断知由[3]的定理5.3.1知，系统(2)的解以M 为极限，M 是的最大不变子集.注意到，当且仅当x=y=z=1 时，即对所有的 t，S(t)=S* ,I(t)=I*,故M={S*,I*} ，说明又由系统(2)的第三个方程有因此E* 是全局渐近稳定的.证毕.参考文献:[1]Takeuchi Y,Ma W,Beretta E.Global asymptotic properties of a delay SIR epidemic model with finite incubation times[J]．Nonlinear Anal，2000,42:931～947.[2]ConnellMcCluskey plete global stability for an SIR epidemic model with delay-Distributed or discrete[J].Nonlinear Analysis:Real World Applications,2010,11:55～59.[3]Hale J,Lunel S V.Introduction to Functional Differentialequations[M].Berlin:Springer-Verlag,1993.[4]Kar T K,Ashim Batabyal.Stability analysis and optimal control of an SIR epidemic model with vaccination[J].BioSystems,2011,104:127～135.[5]廖晓昕.稳定性的数学理论及应用[M]. 武汉:华中师范大学出版社，1988.[6]陈晓英,吴亭.一类具有分布时滞饱和特性发生率的SIR传染病模型[J].福州大学学报(自然科学版),2010,38(6)：808～812.。

[Linear system theory and design］Absolutely integrable 绝对可积Adder 加法器Additivity 可加性Adjoint 伴随Aeronautical航空的Arbitrary 任意的Asymptotic stability渐近稳定Asymptotic tracking 渐近跟踪Balanced realization 平衡实现Basis 基BIBO stability 有界输入有界输出稳定Black box 黑箱Blocking zero 阻塞零点Canonical decomposition 规范分解Canonical规范Capacitor 电容Causality 因果性Cayley-Hamilton theorem 凯莱—哈密顿定理Characteristic polynominal 特征多项式Circumflex 卷积Coefficient 系数Cofactor 余因子Column degree 列次数Column—degree—coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Common Divisor公共因式Companion—form matrix 规范型矩阵Compensator 调节器，补偿器Compensator equation补偿器方程Control configuration 控制构型Controllability 能控性Convolution 卷积Conventional常规的Coprimeness互质Corollary推论Cyclic matrix 循环矩阵Dead beat design 有限拍设计Decoupling 解耦Degree of rational function有理矩阵的次数Description of system系统描述Derivative 导数Determinant 行列式Diagonal对角型Discretization 离散化Disturbance rejectionDivisor 因式Diverge分叉Duality 对偶Eigenvalue特征值Eigenvextor 特征向量Empirical 经验Equivalence 等价Equivalence transformation等价变换Exhaustive 详细的Exponential function 指数函数Extensively广泛地Filter 滤波Finite time有限时间Finite 有限的Fraction 分式Polynomial fraction 多项式分式Fundamental cutest 基本割集Fundamental loop 基本回路Fundamental matrix 基本解阵Gramian 格拉姆Geometric几何的Hankel matrix Hankel 矩阵Hankel singular valueHomogeneity奇次性Hurwitz polynomial Hurwitz 多项式Implementable transfer function 可实现的传递函数Impulse response 脉冲响应Impulse response matrix 脉冲响应矩阵Impulse response sequence 脉冲响应序列Index指数Inductance 自感系数Inductor 电感Integrator 积分器Internal model principle 内模原理Intuitively 直观的Inverse 逆Jacobian 雅可比Jordan block约当块Jordan form matrix 约当型矩阵Lapalace expansion 拉普拉斯展开Lapalace transform 拉普拉斯变换Left multiple 左倍式Linear algebraic equation线性代数方程Linear space 线性空间Linearity 线性性Linearization 线性化Linearly dependent 线性相关Linearly independent 线性无关Lumpedness 集中（参数）性Lyapunov equation Lyapunov方程Lyapunov theorem Lyapunov定理Lyapunov transform Lyapunov变换Markov parameter 马尔科夫参数Magnitude 模Manuscript 手稿，原稿Marginal stability 稳定Matrix 矩阵Minimal polynomial 最小多项式Minimal realization 最小实现Minimum energy control 最小能量控制Minor 子行列式Model matching 模型匹配Model reduction 模型降维Monic 首1的Monic polynomial首1的多项式Multiple 倍式Multiplier 乘法器Nilpotent matrix 幂为零的矩阵Nominal equation 标称(名义)方程Norm 范数Nonsingular非奇异Null space 零空间Nullity 零度(零空间的维数）Observability 能观性Observer 观测器Op—amp circuit 运算发大器Orthogonal 正交的Orthogonal matrix正交矩阵Orthogonality of vectors 向量的正交性Orthonormal set 规范正交集Orthonormalization 规范正交化Oscillator振荡器Parenthesis圆括号Parameterization 参数化Pendulum钟摆Periodic 周期的Periodic state system 周期状态系统Pertinent 相关的Plant被控对象Pole 极点Pole placement 极点配置Pole-zero cancellation 零极点对消Pole—zero excess inequalityPolynomial多项式Polynomial matrix 多项式矩阵Column degree 列次数Column degree 列次数Column—degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Echelon formLeft divisor 左因式Left multiple 左倍式Right divisor 右因式Right multiple 右倍式Row degree 行次数Row-degree-coefficient matrix 行次数系数矩阵Row reduced行既约Unimodular 幺模Positive正的Positive definite matrix正定矩阵Positive semidefinite matrix正定矩阵Power series 幂级数Primary dependent column 主相关列Primary dependent column 主相关行Prime素数的Principal minor 主子行列式Pseudoinverse 伪逆Pseudostate 伪状态QR decomposition QR 分解Quadratic二次的Quadratic form二次型Range space 值域空间Rank秩Rational function 有理函数Biproper双真Improper 不真Proper 真Strictly proper 严格真Rational matrix 有理矩阵Biproper双真Improper 不真Proper 真Strictly proper 严格真Reachability 可达性Realization 状态空间可实现性Balanced 平衡的（实现）Companion form 规范型（实现) Controllable form能控型（实现) Input-normal 输入规范（实现）Minimal 最小(实现)Observable-form能控型（实现）Output—normal 输出规范（实现）Time-varying 时变(实现) Reduced 既约的Regulator problem 调节器问题Relaxedness 松弛Response 响应Impulse 脉冲Zero-input 零输入Zero-state 零状态Remainder 余数Resistor 电阻Resultant 结式Generalized Resultant 广义结式Sylvester Resultant Sylvester（西尔维斯特）结式Right divisor右因子Robust design 鲁棒设计Robust tracking 鲁棒跟踪Row indices 行指数集Saturate 使饱和Scalar 标量Schmidt orthonormalizationSemidefinite 半正定的Similar matrices 相似矩阵Separation property 分离原理Servomechanism 司服机制Similar transformation 相似变换Singular value 奇异值Singular value decomposition奇异值分解Spectrum 谱Stability 稳定Asymptotic stability 渐近稳定BIBO stability 有界输入有界输出稳定Stability in the sense of Lyapunov 李雅普诺夫意义下的稳定Marginal stability 临界稳定Total stability 整体稳定Stabilization镇定(稳定化)State 状态State variable 状态变量State equation 状态方程Discrete—time State equation 离散时间状态方程Discretization State equation 离散化状态方程Equivalent State equation 等价状态方程Periodic State equation 周期状态方程Reduction State equation 状态方程Solution of State equation 状态方程的解Time-varying State equation 时变状态方程State estimator 状态估计器Asymptotic State estimator 渐近状态估计器Closed-loop State estimator 闭环状态估计器Full—dimentional State estimator 全维状态估计器Open—loop State estimator 开环状态估计器Reduced-dimentional State estimator 降维状态估计器State feedback 状态反馈State-space equation 状态空间模型State transition matrix 状态转移矩阵Superposition property 叠加性Sylvester resultant （西尔维斯特）结式Sylvester's inequality （西尔维斯特）不等式System 系统Superposition叠加Terminology 术语Total stability 整体稳定Trace of matrix 矩阵的迹Tracking 跟踪Transfer function 传递函数Discrete Transfer function离散传递函数Pole of Transfer function传递函数的极点Zero of Transfer function传递函数的零点Transfer matrix 传递函数矩阵Blocking zero of matrix 传递函数矩阵的阻塞零点Pole of Transfer function传递函数矩阵的极点Transmission Zero of Transfer function传递函数矩阵的传输零点Transistor 晶体管Transmission zero 传输零点Tree 树Truncation operator 截断算子Trajectory 轨迹Transpose 转置Triangular 三角形的Unimodular matrix 么模阵Unit-time-delay element 单位时间延迟单元Unity—feedback 单位反馈Vandermonde matrix Vandermonde矩阵Vector 向量well posed 适定的well—posedness 适定性Yield 等于,得出Zero 零点Blocking zero 阻塞零点Minimum-phase zero 最小相位零点nonminimum-phase zero 非最小相位零点transmission zero 传输零点zero-input response 零输入响应zero—pole—gain form 零极点增益形式zero—state equivalence零状态等价zero-state response零状态响应z—transform z—变换。

摘要高血压是心脑血管疾病的重要危险因素。

动态血压监测已成为识别和诊断高血压、评估心脑血管疾病风险、评估降压疗效、指导个体化降压治疗不可或缺的检测手段。

本指南对2015年发表的《动态血压监测临床应用专家共识》进行了更新，详细介绍了动态血压计的选择与监测方法、动态血压监测的结果判定与临床应用、动态血压监测的适应证、特殊人群动态血压监测、社区动态血压监测应用以及动态血压监测临床应用展望，旨在指导临床实践中动态血压监测的应用。

关键词 动态血压监测；血压管理；指南；高血压2020 Chinese Hypertension League Guidelines on Ambulatory Blood Pressure MonitoringWriting Group of the 2020 Chinese Hypertension League Guidelines on Ambulatory Blood Pressure Monitoring.Corresponding Author: WANG Jiguang, Email: jiguangwang@2020中国动态血压监测指南中国高血压联盟《动态血压监测指南》委员会指南与共识AbstractHypertension is an important risk factor for cardiovascular and cerebrovascular diseases. Ambulatory blood pressure monitoring (ABPM) has become an indispensable technique for the detection of hypertension, risk assessment of cardiovascular and cerebrovascular diseases, therapeutic monitoring, and guidance of the individualized treatment. Based on the “2015 expert consensus on the clinical use of ambulatory blood pressure monitoring”, the current guideline updates recommendations on the major issues of ABPM, such as the device requirements and methodology, interpretation of the reported results, clinical indications, application on special populations, the utility in the community and future perspectives. It aims to guide the clinical application practice of ABPM in China.Key words ambulatory blood pressure monitoring; blood pressure management; guideline; hypertension(Chinese Circulation Journal, 2021, 36: 313.)高血压是心脑血管疾病的重要危险因素，与心脑血管疾病发病和死亡密切相关[1-3]。

医学免疫英语试题及答案一、选择题（每题2分，共20分）1. Which of the following is not a component of innate immunity?A. Complement systemB. Phagocytic cellsC. Physical barriersD. Antibodies2. The primary function of the spleen in the immune system is:A. To produce antibodiesB. To filter bloodC. To store white blood cellsD. To synthesize cytokines3. What is the process by which the adaptive immune system recognizes and remembers pathogens?A. InflammationB. Immunological memoryC. HypersensitivityD. Autoimmunity4. The major histocompatibility complex (MHC) is involved in:A. Cell adhesionB. Antigen presentationC. Immune cell signalingD. All of the above5. Which type of T cell is responsible for cytotoxic activity?A. Helper T cellsB. Cytotoxic T cellsC. Regulatory T cellsD. Memory T cells二、填空题（每题2分，共10分）6. The first line of defense in the immune system includes physical barriers such as the _______ and the _______.7. The process of an antigen binding to a specific B cell receptor triggers the B cell to _______ and differentiateinto _______ or _______.8. Immunoglobulins, also known as antibodies, are composed of four polypeptide chains: two identical _______ and twoidentical _______.9. The _______ is a type of white blood cell that plays a central role in the adaptive immune response.10. Vaccines are used to prevent infectious diseases by stimulating the production of _______ without causing the disease.三、简答题（每题5分，共20分）11. Describe the role of cytokines in the immune response.12. Explain the concept of antigen presentation and its importance in the adaptive immune system.13. What are the differences between active and passive immunity?14. Discuss the potential risks and benefits of using immunosuppressive drugs in organ transplantation.四、论述题（每题15分，共30分）15. Discuss the mechanisms of the innate and adaptive immune responses and how they complement each other.16. Analyze the role of the immune system in autoimmune diseases and the strategies for their treatment.答案：一、选择题1. D. Antibodies2. B. To filter blood3. B. Immunological memory4. B. Antigen presentation5. B. Cytotoxic T cells二、填空题6. skin, mucous membranes7. proliferation, plasma cells, memory B cells8. heavy chains, light chains9. T lymphocyte10. antibodies三、简答题11. Cytokines are small proteins that play a crucial role in cell signaling and communication during the immune response. They can regulate inflammation, activate immune cells, and stimulate the production of other cytokines.12. Antigen presentation is the process by which antigens are displayed on the surface of antigen-presenting cells (APCs) for recognition by T cells. This is essential for initiatinga specific immune response against a pathogen.13. Active immunity is acquired through exposure to a pathogen or vaccination and leads to the development ofimmunological memory. Passive immunity is temporary and involves the transfer of antibodies from one individual to another, often through injection.14. Immunosuppressive drugs can prevent organ rejection in transplant patients but also increase the risk of infections and malignancies due to their suppressive effect on the immune system.四、论述题15. The innate immune response is the first line of defense and acts quickly to limit the spread of pathogens. It includes physical barriers, chemical barriers, and cellular responses. The adaptive immune response is slower but more specific and can develop immunological memory. Both systems work together to provide a comprehensive defense against pathogens.16. The immune system in autoimmune diseases mistakenly attacks the body's own tissues. Treatment strategies include immunosuppressive drugs to reduce the immune response, targeted therapies to address specific immune cells or molecules, and symptom management to alleviate the effects of the disease.。

乳胶免疫层析法定量检测猪尿液中的盐酸克伦特罗管 笛1,2，魏 颖1,2，王 旗1,2，李俊博1,2，武会娟1,2,*（1.北京市理化分析测试中心，北京 100089；2.北京市基因测序与功能分析工程技术研究中心，北京 100089）摘 要：目的：建立以彩色乳胶微球为标记物的免疫层析技术检测猪尿液中的盐酸克伦特罗（clenbuterol ，CLB ）。

方法：彩色乳胶微球标记CLB 抗体并真空冷冻干燥，盐酸克伦特罗-牛血清白蛋白（CLB-bovine serum albumin ，CLB-BSA ）人工抗原与羊抗鼠抗体分别喷到硝酸纤维素膜上作检测线和质控线，猪尿液样本与冻干微球混匀后，插入试纸条，通过读条仪进行测定。

结果：通过理化参数的优化，吐温-20为最优表面活性剂，选用300 nm 乳胶微球，层析时间为9 min 。

方法的检出限为0.013 ng /mL ，回收率范围在97.8%～106.0%之间，相对标准偏差不高于9.6%。

结论：本研究成功建立了一种灵敏、准确的检测猪尿液中CLB 的彩色乳胶免疫层析方法。

关键词：免疫层析试纸；彩色乳胶；盐酸克伦特罗；猪尿液；定量检测Development of Dyed Latex Bead-Based Immunochromatographic Assay for Quantitative Detection ofClenbuterol in Swine UrineGUAN Di 1,2, WEI Ying 1,2, WANG Qi 1,2, LI Junbo 1,2, WU Huijuan 1,2,*(1. Beijing Center for Physical and Chemical Analysis, Beijing 100089, China;2. Beijing Engineering Technique Research Center for Gene Sequencing & Function Analysis, Beijing100089, China)Abstract: Objective: Aiming at the development of an immunochromatographic method based on dyed latex beads for the detection of clenbuterol (CLB) in swine urine. Methods: Dyed latex beads were conjugated with anti-CLB antibody and vacuum freeze-dried. CLB-bovine serum albumin (CLB-BSA) conjugate and goat anti-mouse IgG antibodies were coated onto nitrocellulose membranes to establish test and control lines, respectively. The test was carried out by placing swine urine into the sample well and mixed with dyed latex beads. After incubation, the optical density was measured by a microplate reader. Results: After optimizing the physical and chemical parameters, the best surfacant was Tween-20, 300 nm dyed latex beads were chosen, and the incubation time was 9 minutes. The limit of detection (LOD) was 0.013 ng /mL. The recoveries from spiked swine urine varied from 97.8% to 106.0%, with relative standard deviation of less than 9.6%. Conclusion: A sensitive and accurate method for the detection of CLB was successfully developed in this paper.Key words: immunochromatographic strip; dyed latex beads; clenbuterol; swine urine; quantitative detection DOI:10.7506/spkx1002-6630-201712042中图分类号：TS207.3 文献标志码：A 文章编号：1002-6630（2017）12-0273-05引文格式：管笛, 魏颖, 王旗, 等. 乳胶免疫层析法定量检测猪尿液中的盐酸克伦特罗[J]. 食品科学, 2017, 38(12): 273-277. DOI:10.7506/spkx1002-6630-201712042. GUAN Di, WEI Ying, WANG Qi, et al. Development of dyed latex bead-based immunochromatographic assay for quantitative detection of clenbuterol in swine urine[J]. Food Science, 2017, 38(12): 273-277. (in Chinese with English abstract) DOI:10.7506/spkx1002-6630-201712042. 收稿日期：2016-07-22基金项目：北京市优秀人才项目（2014400685627G277）；北京市科学技术研究院青年骨干计划项目（201415）作者简介：管笛（1983—），男，副研究员，博士，研究方向为分析化学。

哈工大胡颖组CancerCell揭示肿瘤细胞调控氧化应激的新机制BioArt按：2013年James D. Watson发表的一篇题为“Oxidants, antioxidants and the current incurability of metastatic cancers”的论文，声称“自双螺旋之后我最重要的工作”（among my most important work since the double helix）。

其论文的而核心主要围绕活性氧（ROS）与肿瘤治疗展开，这主要是因为ROS在抗肿瘤中的“两面性”所决定的。

放化疗药物通常激发产生更多的ROS来诱导肿瘤细胞凋亡，然而肿瘤细胞一旦启动抑制ROS的机制，则导致耐药。

因此，研究肿瘤细胞调控氧化应激的分子机制具有重要的意义。

10月12日，哈尔滨工业大学生命科学与技术学院胡颖课题组在Cancer Cell杂志发表题为“iASPP is an antioxidative factor and drives cancer growth and drug resistance by competing with Nrf2 for Keap1 binding”的论文，揭示肿瘤细胞调控ROS的新机制，为逆转肿瘤耐药提供了新的线索。

论文解读：人类细胞不可避免地暴露于来自于外界因素（如UV）以及细胞内有氧代谢所产生的活性氧（reactive oxygen species, ROS），少量ROS是调控细胞正常生理活动的重要信号分子，高水平ROS则会作用于包括DNA在内的生物大分子，致其损伤，破坏其功能，成为肿瘤等疾病发生的重要推动力【1】。

的确，大量研究表明，肿瘤内ROS的水平一般高于同一组织来源的正常对照。

值得注意的是，如果ROS不断积累，超过死亡阈值，则会导致细胞凋亡。

放射治疗以及很多化疗药物均可通过促进ROS过度累积的方式发挥其杀伤肿瘤细胞的作用，而肿瘤细胞一旦启动抑制ROS的机制，则导致耐药【2,3】。

高效溶液法制备延迟荧光电致发光器件研究谢莉莎;王会;孟令强;陈必强;王鹰【摘要】本文以2-[对-N,N-二苯基氨基-苯基]-S-二氧硫杂蒽酮(TXO-TPA)为发光材料,4,4 ',4”-三(9-咔唑基)三苯胺(TCTA)为主体材料,通过溶液法与真空蒸镀相结合的工艺,制备了高效延迟荧光型电致发光器件.为了考察不同电子传输材料对器件性能的影响,分别选取TmPyPB、TPBI、BCP、Alq3作为电子传输层制备器件,并对器件的性能进行系统的研究.结果表明:由于1,3,5-三(1-苯基-1H-苯并咪唑-2-基)苯(TPBI)具有合适的HOMO/LUMO能级、高的电子迁移率以及高的三重态能级,利于电子的传输和激子的阻挡,以其为电子传输层的器件显示出最佳的性能,器件的开启电压低至3.6V,电流效率达到16.2 cd/A,最大的EQE达到5.97％.【期刊名称】《影像科学与光化学》【年(卷),期】2015(033)004【总页数】8页(P292-299)【关键词】溶液法;延迟荧光器件;电子传输层;有机发光二极管【作者】谢莉莎;王会;孟令强;陈必强;王鹰【作者单位】北京化工大学生命科学与技术学院,北京100029;中国科学院理化技术研究所光化学转换功能材料重点实验室,北京100190;中国科学院理化技术研究所光化学转换功能材料重点实验室,北京100190;中国科学院理化技术研究所光化学转换功能材料重点实验室,北京100190;北京化工大学生命科学与技术学院,北京100029;中国科学院理化技术研究所光化学转换功能材料重点实验室,北京100190【正文语种】中文自1987年Tang［1］报道了有机电致发光二极管（Organic Light－Emitting Device，OLED）以来，OLED以其柔性、超薄、自主发光、视角宽、能耗低以及可大面积加工等优点，作为新型的显示照明技术而受到广泛关注。

在早期的荧光发光器件中［2］，由于受电子自旋统计规律的限制，理论上仅有25%的单重态激子可以以辐射跃迁的形式而发光，而其中75%的三重态激子则由于T1→S0的自旋禁阻而无法得到利用，导致荧光OLED器件的内量子效率（Internal Quantum Efficiency，IQE）最高仅为25%。

13Nonlinear DynamicsAn International Journal of Nonlinear Dynamics and Chaos in Engineering SystemsISSN 0924-090X Volume 74Combined 1-2Nonlinear Dyn (2013) 74:133-142DOI 10.1007/s11071-013-0953-1Explicit ultimate bound sets of a newhyperchaotic system and its application in estimating the Hausdorff dimension Pei Wang, Yuhuan Zhang, Shaolin Tan & Li WanYour article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprintis for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further depositthe accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publicationor later and provided acknowledgement is given to the original source of publicationand a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at ”.13Nonlinear Dyn(2013)74:133–142DOI10.1007/s11071-013-0953-1O R I G I NA L PA P E RExplicit ultimate bound sets of a new hyperchaotic system and its application in estimating the Hausdorff dimension Pei Wang·Yuhuan Zhang·Shaolin Tan·Li WanReceived:30March2013/Accepted:7May2013/Published online:21May2013©Springer Science+Business Media Dordrecht2013Abstract Ultimate bound estimation of chaotic sys-tems is a difficult yet interesting mathematical ques-tion.At present,explicit ultimate bound sets can be analytically obtained only for some special chaotic systems,and few results are known for hyper-chaotic ones.In this paper,through the Lagrange multiplier method and set operations,one derives two kinds of explicit ultimate bound sets for a novel hyperchaotic system.Based on the estimated result and optimiza-tion method,one further estimates the Hausdorff di-mension of the hyperchaotic attractor.Numerical sim-ulations show the effectiveness and correctness of the conclusions.The investigations enrich the related re-sults for the hyperchaotic systems,and provide sup-port for the assertion:hyperchaotic systems are ulti-mately bounded.P.Wang( )·Y.ZhangSchool of Mathematics and Information Sciences,Henan University,Kaifeng475004,Chinae-mail:wp0307@S.TanLSC,Institute of Systems Science,Academyof Mathematics and Systems Science,Chinese Academyof Sciences,Beijing100190,P.R.ChinaL.WanSchool of Mathematics and Computer Science,Wuhan Textile University,Wuhan430073,P.R.China Keywords Hyperchaotic system·Ultimate bound·Lagrange multiplier method·Optimization·Hausdorff dimension1IntroductionChaotic systems are ultimately bounded[1–13].In that,the phase portraits of the systems will be ulti-mately trapped in some compact sets[4,9,10].Ul-timate bound estimation of chaotic systems is a fun-damental mathematical problem.However,it is diffi-cult to obtain explicit ultimate bound sets for chaotic systems,especially for high dimensional ones.There are two main difficulties.On the one hand,it is fun-damentally difficult tofind an appropriate Lyapunov function[10].On the other hand,if one couldfind an appropriate Lyapunov function for a certain chaotic system,it is still difficult to derive the explicit ultimate bound sets for the chaotic system,especially for high dimensional ones.Following,one briefly overviews some recent pro-gresses on the ultimate bound estimation of the hy-perchaotic systems.In[8],the authors prove that the hyper-chaotic Lorenz–Haken system is ultimately bounded,but no explicit bound sets were derived due to the complexity of the optimization problem. And in[9],we investigate the ultimate boundness of the hyperchaotic Lorenz–Stenflo system,through the method of dimension reduction,we have obtained some explicit bound sets for the hyperchaotic system.134P.Wang et al.Recently,based on the optimization method and ma-trix theory,we have investigated the ultimate bound-ness of the general quadratic autonomous dynamical systems,and derive a sufficient condition for the gen-eral system.In addition,a class of high dimensional quadratic autonomous chaotic systems are proved to be ultimately bounded[10].Though much works have been done,and one couldfind some appropriate Lya-punov functions for some systems by following[10], it is still technically difficult to obtain explicit bound sets for the high dimensional chaotic systems[10].For example,in[10],if one wants to derive explicit ulti-mate bound sets for the high dimensional dynamical systems,one has to solve some high dimensional op-timization problems.However,the high dimensional optimization problems are always the main obsta-cle.Ultimate bound sets have important applications in chaos control and its synchronization.It can also be applied in estimating the fractal dimensions of chaotic attractors,such as the Hausdorff dimension and the Lyapunov dimension of chaotic attractors[12,14–18]. The Hausdorff dimension of some3D chaotic attrac-tors have been estimated[12,14–18].It is intriguing to estimate the Hausdorff dimension of the hyperchaotic attractors by using their ultimate bound sets.Motivated by the above two issues,based on the optimization method and the Lagrange multiplier method,we show a possible way to explore the ul-timate bound sets for hyperchaotic systems[13].And discuss the application of the ultimate bound sets in the estimation the Hausdorff dimension of the hyper-chaotic attractors.The rest of the paper is organized as follows.Section2investigates the ultimate bound sets for the new hyperchaotic system.The applications of the ultimate bound sets in the estimation of the Haus-dorff dimension are discussed in Sect.3.Concluding remarks are presented in Sect.4.2Ultimate bound estimation of the new hyperchaotic systemFirst of all,we introduce the existing general method for bound estimation.A general method of ultimate bound estimation has been presented in our previous paper[10],the main conclusion is as follows.The general quadratic autonomous dynamical sys-tem(GQADS)[9,10,19]is described by ˙X=AX+ni=1x i B i X+C,(1)where X=(x1,x2,...,x n)T∈R n is the state vec-tor,A=(a ij)n×n∈R n×n,B i=(b ijk)n×n∈R n×n, and C=(c1,c2,...,c n)T∈R n.Also,all elements of B1,B2,...,B n satisfy b kij=b jik for i,j,k= 1,2,...,n.System(1)contains all quadratic au-tonomous chaotic systems as its special cases,such as the well-known Lorenz system,the Chen system, the unified system,the Lüsystem,the Rössler system [19,21–23].Following,P>0means that matrix P is positive definite.Similarly,P<0is negative definite.A suf-ficient condition obtained in[10]for system(1)is as follows.Lemma1[10]Suppose that there exists a real sym-metric matrix P>0and a vectorμ∈R n such thatQ=A T P+P A+2B T1PμT,B T2PμT,...,B T n PμTT<0(2)andni=1x i X TB T i P+P B iX=0(3)for any X=(x1,x2,...,x n)T∈R n,then system(1)is bounded and has the following ultimate bound set:Ω=X∈R n|(X+μ)T P(X+μ)≤R max,(4) where R max is a real number to be determined by max(X+μ)T P(X+μ)s.t.X T QX+2μT P A+C T PX+2C T Pμ=0.(5)Lemma1only gives a sufficient condition.In[10], boundness of some4D,6D,or even9D chaotic sys-tems have been discussed,but no explicit ultimate bound sets have been obtained for these high dimen-sional chaotic system.In real world,the optimization problem(5)is very difficult to be analytically solved. In the following,taken a newly proposed hyperchaotic system as an example,we introduce a possible way to explore the explicit ultimate sets of the high dimen-sional chaotic systems.Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating135 Recently,a new hyperchaotic system is proposedin[13],which is described as⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩˙x1=a(x2−x1),˙x2=bx1−x1x3−cx2+x4,˙x3=x1x2−dx3,˙x4=−kx2−rx4.(6)When a=12,b=23,c=1,d=2.1,k=6,r=0.2, system(6)is hyperchaotic.Rewrite system(6)in the form of system(1),one hasA=⎡⎢⎢⎣−a a00b−c0100−d00−k0−r⎤⎥⎥⎦,B1=⎡⎢⎢⎣0000 00−1/20 01/200 0000⎤⎥⎥⎦,B2=⎡⎢⎢⎣000000001/20000000⎤⎥⎥⎦,B3=⎡⎢⎢⎣0000−1/200000000000⎤⎥⎥⎦,B4=04×4,C=04×1.Suppose p11>0,p22>0,and takeP=⎡⎢⎢⎣p110000p220000p2200001kp22⎤⎥⎥⎦,μ=0,0,−ap11+bp22p22,0T.Then P,μsatisfy Eqs.(2)and(3)in Lemma1.And one hasQ=⎡⎢⎢⎣ap110000cp220000dp220000rkp22⎤⎥⎥⎦,M=0,0,d(ap11+bp22),0T.By Lemma1and the following techniques to solvethe high dimensional optimization problem,one canderive the following proposition for system(6).Proposition1Suppose that a>0,c>0,d>0,r>0,p ii>0,denoteΩ=X∈R4|p11x21+p22x22+p22x3−ap11+bp22p222+p22kx24≤R max.(7)ThenΩis an ultimate bound set of system(6),whereR max=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩d2(ap11+bp22)24p22a(d−a),0<a<min{c,r},d≥max{a+c,a+r};d2(ap11+bp22)24p22c(d−c),0<c<min{a,r},d≥max{a+c,c+r};d2(ap11+bp22)24p22r(d−r),0<r<min{a,c},d≥max{a+r,c+r};(ap11+bp22)2p22,otherwise.Proof From the Lemma1,R max can be determined bysolving the following optimization problem:max V=p11x21+p22x22+p22x3−ap11+bp22p222+p22kx24s.t.ap11x21+cp22x22+dp22x3−ap11+bp222p222+rp22kx24=(ap11+bp22)2d4p22.(8)Letθ=(ap11+bp22)/(2√p22),√p11x1=y1,√p22x2=y2,√p22x3−2θ=y3,√p22/kx4=y4,then the optimization problem(8)is equivalent toproblem:max H=y21+y22+y23+y24s.t.ay21+cy22+d(y3+θ)2+ry24=dθ2.(9)136P.Wang et al.By the Lagrange multiplier method,define:L=y21+y22+y23+y24−ωay21+cy22+d(y3+θ)2+ry24−dθ2,(10)and let:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩∂L∂y1=2y1(1−ωa)=0,∂L∂y2=2y2(1−ωc)=0,∂L∂y3=2y3(1−ωd)−2ωdθ=0,∂L∂y4=2y4(1−ωr)=0,∂L∂ω=−(ay21+cy22+d(y3+θ)2+ry24−dθ2)=0.(11)(i)Whenω=1/a,and d≥2a,a>0,one derivestwo stationary points:y∗1,y∗2,y∗3,y∗4=±dθ√d−2a(d−a)√a,0,dθa−d,0,andH1=Hy∗1,y∗2,y∗3,y∗4=d2(ap11+bp22)24p22a(d−a).(ii)Whenω=1/c,and d≥2c,c>0,one hasy∗1,y∗2,y∗3,y∗4=0,±dθ√d−2c(d−c)√c,dθc−d,0,andH2=Hy∗1,y∗2,y∗3,y∗4=d2(ap11+bp22)24p22c(d−c).(iii)Whenω=1/r,and d≥2r,r>0,one obtainsy∗1,y∗2,y∗3,y∗4=0,0,dθr−d,±dθ√d−2r(d−r)√r,andH3=Hy∗1,y∗2,y∗3,y∗4=d2(ap11+bp22)24p22r(d−r).(iv)Whenω=1/a,ω=1/c,ω=1/r,one hasy∗1,y∗2,y∗3,y∗4=(0,0,−2θ,0)ory∗1,y∗2,y∗3,y∗4=(0,0,0,0),and H4=4θ2or H5=0.Summarizing(i)–(iv),since the constrained con-dition in Eq.(9)is an ellipsoid,the objective func-tion H can achieve its minimum and maximumvalues.Obviously,H5=0is the minimum value of H.Denote the maximum values of H as H max,then,for0<a<min{c,r},d≥max{a+c,a+r},H max=H1=max{H1,H2,...,H5}.For0<c< min{a,r},d≥max{a+c,c+r},H max=H2= max{H1,H2,...,H5}.For0<r<min{a,c},d≥max{a+r,c+r},H max=H3=max{H1,H2, ...,H5}.Otherwise,H max=H4=max{H1,H2, ...,H5}.For simplicity,one denotes S1={(a,c, d,r)∈R4+|0<a<min{c,r},d≥max{a+c, a+r}},S2={(a,c,d,r)∈R4+|0<c<min{a,r}, d≥max{a+c,c+r}},S3={(a,c,d,r)∈R4+|0<r<min{a,c},d≥max{a+r,c+r}},S4=R4+− 3i=1S i.Thus,one hasH max=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩d2θ2a(d−a),(a,c,d,r)∈S1;d2θ2c(d−c),(a,c,d,r)∈S2;d2θ2r(d−r),(a,c,d,r)∈S3;4θ2,(a,c,d,r)∈S4.(12) Substitutedθfor(ap11+bp22)/(2√p22)in H max, and noted that V max=H max=R max,the proof is thus completed. Remark1The above discussions for the hyperchaotic system can be extended to other systems.It is noted that,in work[13],the authors also discussed the boundness of this hyperchaotic system.However,the estimated results only hold for a>1,c>1,d>1, and the estimated results are very rough and also too large.For example,when a=12,b=23,c=1, d=2.1,k=5,r=0.2,a bound estimated in[13]isx21+x22+(x3−35)2+0.2x24≤6138.9,whereas ourresult shows that x21+x22+(x3−35)2+0.2x24≤1225, which is obviously more accurate and conservative than the result in[13].Moreover,our results hold for any a>0,c>0,d>0,r>0.Remark2In work[10],we only demonstrate that some high dimensional chaotic systems are ultimately bounded and have ellipsoidal bound sets;no explicit bound sets have been analytically calculated for4D and higher dimensional dynamical systems.Here,for the new hyperchaotic system,one derives an explicit ultimate bound set through the Lagrange multiplierExplicit ultimate bound sets of a new hyperchaotic system and its application in estimating137 Fig.1The phase portraits and ultimate bound of system(6)in different3D projection planes,where a=12,b=23,c=1,d=2.1,k=5,r=0.2,p11=p22=1method.As far as we know,as to high dimensionalchaotic systems,there are only few reports with ex-plicit bound estimation in existing references[9].To verify the correctness of the conclusion inProposition1,one sets a=12,b=23,c=1,d=2.1,k=5,r=0.2,p11=p22=1,then R max=(ap11+bp22)2/p22,the phase portraits and ultimatebound of the system(6)in different3D projectionplanes are shown in Fig.1,where all phase por-traits are ultimately trapped by the estimated ellip-soid.Figure2shows the case for a=12,b=23,c=1,d=15,k=5,r=0.2,p11=p22=1,whereR max=d2(ap11+bp22)2/[4p22r(d−r)].Here,thesolution of system(6)is a periodic orbit,and the peri-odic orbit is also encircled by the estimated result.The estimated result in Proposition1is a set withfree parameters p11,p22,based on the set operations,one can further derive a more conservative boundaryfor each state variable in the system(6).Corollary1Suppose that a>0,c>0,d>0,r>0,then based on Proposition1,the smallest range foreach state variable in the system(6)is given as fol-lows:|x1|≤⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩√bd√d−a,(a,c,d,r)∈S1;√abd√c(d−c),(a,c,d,r)∈S2;√abd√r(d−r),(a,c,d,r)∈S3;2√(a,c,d,r)∈S4;(13)|x2|≤⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩bd2√a(d−a),(a,c,d,r)∈S1;bd2√c(d−c),(a,c,d,r)∈S2;bd2√r(d−r),(a,c,d,r)∈S3;b,(a,c,d,r)∈S4;(14)|x3−b|≤⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩bd2√a(d−a),(a,c,d,r)∈S1;bd2√c(d−c),(a,c,d,r)∈S2;bd2√r(d−r),(a,c,d,r)∈S3;b,(a,c,d,r)∈S4;(15)138P.Wang etal. Fig.2The phase portraits and ultimate bound of system(6)in different3D projection planes,where a=12,b=23,c=1,d=15,k=5,r=0.2,p11=p22=1|x4|≤⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩bd√k2√a(d−a),(a,c,d,r)∈S1;bd√k2√c(d−c),(a,c,d,r)∈S2;bd√k2√r(d−r),(a,c,d,r)∈S3;b√k,(a,c,d,r)∈S4.(16)Proof From Proposition1,we also denoteθ=(ap11+bp22)/(2√p22),then one has|x1|≤R max/p11,|x2|≤R max/p22,|x3−2θ/√p22|≤R max/p22,|x4|≤max22whereR max=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩d2θ2a(d−a),(a,c,d,r)∈S1;d2θ2c(d−c),(a,c,d,r)∈S2;d2θ2r(d−r),(a,c,d,r)∈S3;4θ2,(a,c,d,r)∈S4.(17)Therefore,tofind the upper bound for each state vari-able,it is sufficient tofind the lower bound ofθ/√p11andθ/√p22.Since p11,p22are arbitrary positive numbers,andθ√p11=a2√p11√p22+b2√p22√p11≥√ab,therefore,one can obtain the bound for x1as shown inEq.(13).For the bound of x2,x3,x4,there is a same term:θ√p22=12(εa+b),whereε=p11/p22,which can also be arbitrary.Tak-ingε=1/N(N=1,2,...),and noted that the inter-sections of ultimate bound sets are also an ultimatebound sets[2].DenoteR maxp22=121Na+bc0,Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating 139Table 1Comparison between the differentestimated results and actual bound for the system (6),where a =12,b =23,c =1,d =2.1,k =5,r =0.2,p 11=p 22=1Variables Proposition 1Corollary 1Actual bound x 1(−35.00,35.00)(−33.23,33.23)(−15.85,16.57)x 2(−35.00,35.00)(−23.00,23.00)(−19.81,20.89)x 3(0.00,70.00)(0.00,46.00)(1.91,38.87)x 4(−78.26,78.26)(−51.43,51.43)(−39.20,38.26)where c 0is a constant,which can be easily derived from the expression of R max asc 0=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩d √a(d −a),(a,c,d,r)∈S 1;d √c(d −c),(a,c,d,r)∈S 2;d √r(d −r),(a,c,d,r)∈S 3;2,(a,c,d,r)∈S 4.(18)Since∞ N =1x 2∈R ||x 2|≤12 1Na +bc 0= x 2∈R ||x 2|≤12bc 0,one can derive a smaller bound set for x 2as shown in Eq.(14).Similarly,one obtains more conservative bounds for x 3and x 4as shown in Eqs.(15)–(16).The proof is thus completed. For a =12,b =23,c =1,d =2.1,k =5,r =0.2,p 11=p 22=1,Table 1shows the estimated bound-ary for each state variable in Proposition 1and Corol-lary 1,as well as the actual bound from numerical so-lution of the system (6).Obviously,the estimated re-sult in Corollary 1is more conservative than Proposi-tion 1.And the estimated ranges all contain the actual bound of each state variable,which demonstrate the correctness and advantage of Corollary 1.It is noted that the bound for each state variable in Corollary 1is only determined by the system param-eters.Therefore,it is very convenient to be applied in chaos control and synchronization.3Application in the estimation of the Hausdorff dimensionIn this section,one discusses the application of ulti-mate bound estimation in estimating the Hausdorff di-mension of the hyperchaotic attractor.The Hausdorffdimension is a very important characteristic quantity for chaotic systems,which plays the role as period and amplitude to an oscillator.Denote J as the Jacobian of the hyperchaotic sys-tem (6).For some positive definite matrix Θ,suppose λ1≥λ2≥λ3≥λ4to be solutions of the following characteristic equation:det ΘJ +J T Θ−λΘ=0,(19)where λi are functions of x 1,x 2,x 3,x 4,and Θis inde-pendent of the state vector.From the works of Leonov [14–17]and Pogromsky et al.[12,18],one has the following lemma.Lemma 2[12,14,17,18]Suppose that for some posi-tive definite matrix Θ,there exists a number σ∈[0,1),such thatλ1(x 1,x 2,x 3,x 4)+λ2(x 1,x 2,x 3,x 4)+λ3(x 1,x 2,x 3,x 4)+σλ4(x 1,x 2,x 3,x 4)<0(20)for all x 1,x 2,x 3,x 4in some trapping region ,then the Hausdorff dimension dim H K of the hyperchaotic at-tractor is upper bounded by 3+σ.Remark 3In existing works [12,14,17,18],re-searchers mainly considers the Hausdorff dimension estimation for 3D chaotic systems.To the best of our knowledge,Hausdorff dimension estimation for hy-perchaotic systems are rarely investigated,which is mainly because explicit ultimate bound sets for hyper-chaotic systems could not be easily obtained.Following,by using Lemma 2,for a >0,c >0,d >0,r >0,one estimates the upper bound of the Hausdorff dimension of the hyperchaotic system (6).The Jacobian of the hyperchaotic system isJ =⎡⎢⎢⎣−a a 00b −x 3−c −x 11x 2x 1−d 00−k 0−r⎤⎥⎥⎦.140P.Wang et al.Taking Θ=diag {1,1,1,1/k },and from (19),λis the eigenvalue of matrix ΘJ Θ−1+J T=⎡⎢⎢⎣−2a a +b −x 3x 20a +b −x 3−2c 00x 20−2d 0000−2r⎤⎥⎥⎦.Therefore,λ1(X)+λ2(X)+λ3(X)+λ4(X)=−2(a +c +d +r)<0.(21)It follows that λ4(X)<0for any X =(x 1,x 2,x 3,x 4)T .From (20)and (21),one has σ>1−2(a +c +d +r)|λ4(X)|.(22)Thus,to find the upper estimation of dim H K ,it is sufficient to find the lower bound λmin 4of λ4(X).Then the upper bound of the Hausdorff dimension is dim H K ≤4−2(a +c +d +r)|λmin 4|.(23)Following,one estimates λmin 4.The characteristicequation from (19)is(λ+2r)(λ+2a)(λ+2c)(λ+2d)−(λ+2c)x 22−(λ+2d)(a +b −x 3)2 =0.(24)It follows that λ=−2r or (λ+2a)(λ+2c)(λ+2d)−(λ+2c)x 22−(λ+2d)(a +b −x 3)2=0.(25)Letη1=2(a +c +d),η2=4(ac +ad +cd)− x 22+(a +b −x 3)2 ,η3=8acd −2 cx 22+d(a +b −x 3)2 .Then,Eq.(25)degenerates into λ3+η1λ2+η2λ+η3=0.(26)Denote R 1=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩d 2(a +b)24a(d −a),(a,c,d,r)∈S 1;d 2(a +b)24c(d −c),(a,c,d,r)∈S 2;d 2(a +b)24r(d −r),(a,c,d,r)∈S 3;(a +b)2,(a,c,d,r)∈S 4.Then,from Proposition 1,set p 11=1,p 22=1,onederives 0≤x 22+(a +b −x 3)2≤R 1.Therefore,4(ac +ad +cd)−R 1≤η2≤4(ac +ad +cd).Further denote β=max {c,d },one obtains 8acd −2βR 1≤η3≤8acd .Thus,one can derive the minimum value of λin Eq.(26)from the following optimization problem:min λs.t.λ3+η1λ2+η2λ+η3=0;η1=2(a +c +d);4(ac +ad +cd)−R 1≤η2≤4(ac +ad +cd);8acd −2βR 1≤η3≤8acd.(27)For example,for a =12,b =23,c =1,d =2.10,k =6,r =0.20,one gets η1=30.20,−1067.80≤η2≤157.20,−4943.40≤η3≤201.60.One obtains λ=−51.15from optimization problem (27).There-fore,λmin 4=min {−51.15,−2r }=−51.15,and dim H K ≤4−2(a +c +d +r)|λ4|≈3.40.That is,for a =12,b =23,c =1,d =2.10,k =6,r =0.20,the upper bound of the Hausdorff dimen-sion of the hyperchaotic attractor is 3.40.For a =12,b =23,c =1,d =2.10,k =6,r =0.20,through the time series of the system,one can also compute the upper bound of the Hausdorff di-mension of the hyperchaotic attractor.For example,from time series of x 2(t)and x 3(t),one can derive −166.83≤η2≤156.99,−363.96≤η3≤200.88.And from the optimization problem,which is similar to (27),one can derive λmin 4=−35.11.Thus,one has dim H K ≤pared with the result by using the estimated ultimate bound set,the estimated result dim H K ≤3.40is very close to dim H K ≤3.13.This demonstrates the correctness of our result.It is noted that one has to numerically solve the differential equa-tions by the time series method,which is time consum-Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating141ing and highly parameter-dependent,therefore,the ad-vantage of ultimate bound estimation in the estimation of the Hausdorff dimension is remarkable.4ConclusionsThis paper has investigated the ultimate bound es-timation of a newly proposed hyper-chaotic system. We introduce a possible way to derive explicit ulti-mate bound sets for high dimensional chaotic systems. One obtains an explicit ellipsoidal bound for such sys-tem.And based on the ellipsoidal bound set and set operations,one further obtains a more conservative boundary for each variable in the hyperchaotic system, which only relies on the system parameters.Numeri-cal simulations show the correctness of our results.The investigations provide support for the assertion that:chaotic systems are ultimately bounded[1–13]. Ultimate bounds of chaotic systems can be applied in chaos control and synchronization,which has been ex-tensively investigated[12,20].It can also be used to estimate the Hausdorff dimension and Lyapunov di-mension of the strange attractor.Based on the esti-mated bound set,one concludes that the Hausdorff dimension of the hyperchaotic attractor is not higher than3.40.At present,ultimate bound estimations of many chaotic systems are still a difficult mathematical prob-lem,such as the well-known Chen system[21],Lüsystem[11,22,23]and multi-scroll chaotic attractors [24,25].How tofind explicit ultimate bound sets for these systems is still an unsolved question.It is also intriguing to explore other possible applications of ul-timate bound estimations,and estimating the ultimate bound for other dynamical systems,such as stochastic neural networks[26,27].Acknowledgements This work was supported by the Science Foundation of Henan University under Grants2012YBZR007, and also partly supported by the National Natural Science Foun-dation of China under Grants60804039,60974081,11271295, and60821091.This work was also partly supported by the Sci-ence and Technology Research Projects of Hubei Provincial De-partment of Education under Grants:D2*******. References1.Leonov,G.,Bunin,A.,Koksch,N.:Attractor localizationof the Lorenz system.Z.Angew.Math.Mech.67,649–656 (1987)2.Krishchenko,A.:Localization of invariant compact sets ofdynamical systems.Differ.Equ.41,1669–1676(2005) 3.Yu,P.,Liao,X.:On the study of globally exponentially at-tractive set of a general chaotic mun.Nonlin-ear Sci.Numer.Simul.13,1495–1507(2008)4.Li,D.,Lu,J.,Wu,X.,Chen,G.:Estimating the ultimatebound and positively invariant set for the Lorenz system and a unified chaotic system.J.Math.Anal.Appl.323, 844–853(2006)5.Liao,X.,Fu,Y.,Xie,S.,Yu,P.:Globally exponentially at-tractive sets of the family of Lorenz systems.Sci.China, Ser.F51,283–292(2008)6.Kanatnikov,A.,Krishchenko,A.:Localization of invariantcompact sets of nonautonomous systems.Differ.Equ.45, 46–52(2009)7.Wang,P.,Li,D.,Wu,X.,Lü,J.:Estimating the ultimatebound for the generalized quadratic autonomous chaotic systems.In:Proc.of the29th Chin.Contr.Conf.,Beijing, China,July29–31,pp.791–795(2010)8.Li,D.,Wu,X.,Lu,J.:Estimating the ultimate bound andpositively invariant set for the hyperchaotic Lorenz–Haken system.Chaos Solitons Fractals39,1290–1296(2009) 9.Wang,P.,Li,D.,Hu,Q.:Bounds of the hyper-chaoticLorenz–Stenflo mun.Nonlinear Sci.Numer.Simul.15,2514–2520(2010)10.Wang,P.,Li,D.,Wu,X.,Lü,J.,Yu,X.:Ultimate boundestimation of a class of high dimensional quadratic au-tonomous dynamical systems.Int.J.Bifurc.Chaos Appl.Sci.Eng.21,2679–2694(2011)11.Zhang,F.,Mu,C.,Li,X.:On the boundness of some solu-tions of the Lüsystem.Int.J.Bifurc.Chaos Appl.Sci.Eng.22,1250015(2012)12.Pogromsky,A.,Santoboni,G.,Nijmeijer,H.:An ultimatebound on the trajectories of the Lorenz system and its ap-plications.Nonlinearity16,1597–1605(2003)13.Li,Y.,Liu,X.,Chen,G.,Liao,X.:A new hyperchaoticLorenz-type system:generation,analysis,and implemen-tation.Int.J.Circuit Theory Appl.39,865–879(2011) 14.Leonov,G.:Lyapunov dimension formulas for Henon andLorenz attractors.St.Petersburg Math.J.13,1–12(2001) 15.Leonov,G.:Lyapunov functions in the attractors dimensiontheory.J.Appl.Math.Mech.76,129–141(2012)16.Leonov,G.:Upper estimates of the Hausdorff dimension ofattractors.Vestn.St.Petersbg.Univ.Ser.1,19–22(1998) 17.Leonov,G.:Strange Attractors and Classical Stability The-ory.St.Petersburg University Press,St.Petersburg(2009) 18.Pogromsky,A.,Nijmeijer,H.:On estimates of the Haus-dorff dimension of invariant compact sets.Nonlinearity13, 927–945(2000)19.Lü,J.,Chen,G.,Cheng,D.:A new chaotic system andbeyond:the generalized Lorenz-like system.Int.J.Bifurc.Chaos Appl.Sci.Eng.14,1507–1537(2004)20.Li,D.,Wang,P.,Lu,J.:Some synchronization strategiesfor a four-scroll chaotic system.Chaos Solitons Fractals42, 2553–2559(2009)21.Chen,G.,Ueta,T.:Yet another chaotic attractor.Int.J.Bi-furc.Chaos Appl.Sci.Eng.9,1465–1466(1999)22.Lü,J.,Chen,G.:A new chaotic attractor coined.Int.J.Bi-furc.Chaos Appl.Sci.Eng.12,659–661(2002)23.Lü,J.,Chen,G.,Cheng,D.,C˘e likovský,S.:Bridge the gapbetween the Lorenz system and the Chen system.Int.J.Bi-furc.Chaos Appl.Sci.Eng.12,2917–2926(2002)。