2007-Smooth second-order sliding modes-missiles guidance application

一种新型永磁同步电机高阶滑模转速观测器的研究纪科辉;周勇;鲁文其【摘要】针对面贴式永磁同步电机无位置传感器速度控制系统存在的速度检测延迟、检测误差较大、动态响应不够快等问题,对永磁同步电机无位置传感器速度控制系统模型和传统滑模速度观测器结构和原理进行了分析,对高阶滑模算法理论进行了推导,结合永磁同步电机数学模型,提出了一种以位置信号为滑模变量,以高阶滑模变结构算法为基础的新型滑模转速观测器,利用Simulink软件构建了基于新型滑模速度观测器永磁同步电机矢量控制系统,对电机的起动特性、速度跟踪特性、抗外界干扰性能进行了研究.研究结果表明,由于新的滑模观测器将转子位置与反电势信号的关系进行了分离,消除了反电势信号处理滤波器对速度估算的延迟,提高了速度检测精度,从而改善了系统的稳态和动态性能.【期刊名称】《机电工程》【年(卷),期】2016(033)009【总页数】5页(P1135-1139)【关键词】高阶滑模;永磁同步电机;无位置传感器控制;转速观测器【作者】纪科辉;周勇;鲁文其【作者单位】浙江理工大学机械与自动控制学院,浙江杭州310018;浙江华丰电动工具有限公司,浙江金华321037;宁波罗杰克智能科技有限公司,浙江宁波315000;浙江理工大学机械与自动控制学院,浙江杭州310018【正文语种】中文【中图分类】TM351永磁同步电机是一个多变量、强耦合的非线性系统，存在参数时变、负载扰动等不确定因素，对控制系统的要求很高，其控制性能的好坏取决于系统对未知状态量的估测精度和动态响应速度，因此研究如何快速而又准确地获取转速和转角信息是研究PMSM无位置传感器控制系统的关键所在。

滑模观测器源于滑模变结构控制方法，具有类似于滑模变结构算法的特点，因而对于系统的不确定性和外部干扰具有较强的鲁棒性和抗扰性[1-2]。

研究表明通过准确设计滑模参数，滑模观测器能够较精确地估计系统参数[3-7]。

目前，在基于滑模算法的电动机无位置传感器转速控制研究方向上，已经有了一定的研究成果。

Automatica38(2002)2159–2167/locate/automaticaBrief PaperNon-singular terminal sliding mode control of rigid manipulatorsYong Feng a,Xinghuo Yu b;∗,Zhihong Man ca Department of Electrical Engineering,Harbin Institute of Technology,Harbin150006,People’s Republic of Chinab School of Electrical and Computer Engineering,Royal Melbourne Institute of Technology University,GPO Box2476V Melbourne,Vic.3001,Australiac School of Computer Engineering,Nanyang Technological University,SingaporeReceived26June2001;received in revised form16June2002;accepted9July2002AbstractThis paper presents a global non-singular terminal sliding mode controller for rigid manipulators.A new terminal sliding mode manifold isÿrst proposed for the second-order system to enable the elimination of the singularity problem associated with conventional terminal sliding mode control.The time taken to reach the equilibrium point from any initial state is guaranteed to beÿnite time.The proposed terminal sliding mode controller is then applied to the control of n-link rigid manipulators.Simulation results are presented to validate the analysis.?2002Elsevier Science Ltd.All rights reserved.Keywords:Terminal sliding mode control;Singularity;Robotic manipulator;Robust control;Lyapunov stability1.IntroductionVariable structure systems(VSS)are well known for their robustness to system parameter variations and external disturbances(Slotine&Li,1991;Utkin,1992; Yurl&James,1988).VSS have been widely used in many applications,such as robots,aircrafts,DC and AC motors, power systems,process control and so on.An aspect of VSS that is of particular interest is the sliding mode control,which is designed to drive and constrain the system states to lie within a neighborhood of the pre-scribed switching manifolds that exhibit desired dynam-ics.When in the sliding mode,the closed-loopresp onse becomes totally insensitive to both internal parameter un-certainties and external disturbances.A characteristic of conventional VSS is that the convergence of the system states to the equilibrium point is usually asymptotical due to the asymptotical convergence of the linear switching manifolds that are commonly chosen.Recently,a terminal sliding mode(TSM)controller was developed(Man&Yu,1997;Yu&Man,1996;Wu,Yu,& This paper was not presented at any IFAC meeting.This paper was recommended for publication in revised form by Associate Editor Jurek Z.Sasiadek under the direction of Editor Mituhiko Araki.∗Corresponding author.E-mail addresses:yfeng@(Y.Feng),x.yu@.au(X.Yu).Man,1998).TSM has been used in the control of rigid ma-nipulators(Man et al.,1994;Tang,1998).The TSM con-cept is related to theÿnite time control(Haimo,1986; Bhat&Bernstein,1997).Compared with linear hyperplane-based sliding modes,TSM o ers some superior properties such as fast,ÿnite time convergence.This controller is par-ticularly useful for high precision control as it speeds up the rate of convergence near an equilibrium point.However,the existing TSM controller design methods still have a singu-larity problem.An initial discussion to avoid the singularity in TSM control systems was presented(Wu et al.,1998). In this paper,a global non-singular terminal sliding mode (NTSM)controller is presented for a class of nonlinear dy-namical systems with parameter uncertainties and external disturbances.A new NTSM manifold is proposed to over-come the singularity problem.The time taken to reach the manifold from any initial state and the time taken to reach the equilibrium point in the sliding mode can be guaran-teed to beÿnite time.The proposed NTSM controller is then applied to the control of n-degree-of-freedom rigid ma-nipulators.Simulation results are presented to validate the analysis.2.Conventional terminal sliding mode controlThe basic principle of TSM control can be brie y sum-marized as follows:consider a second-order uncertain0005-1098/02/$-see front matter?2002Elsevier Science Ltd.All rights reserved. PII:S0005-1098(02)00147-42160Y.Feng et al./Automatica 38(2002)2159–2167nonlinear dynamical system ˙x 1=x 2;˙x 2=f (x )+g (x )+b (x )u;(1)where x =[x 1;x 2]T is the system state vector,f (x )and b (x )=0are smooth nonlinear functions of x ,and g (x )represents the uncertainties and disturbances satisfying g (x ) 6l g where l g ¿0,and u is the scalar control in-put.The conventional TSM is described by the following ÿrst-order terminal sliding variables =x 2+ÿx q=p1;(2)where ÿ0is a design constant,and p and q are positive odd integers,which satisfy the following condition:p ¿q:(3)The su cient condition for the existence of TSM is 12d d ts 2¡−Á|s |;(4)where Á¿0is a constant.For system (1),a commonly used control design isu =−b −1(x ) f (x )+ÿq px q=p −11x 2+(l g +Á)sgn(s );(5)which ensures that TSM occurs.It is clear that if s (0)=0,the system states will reach the sliding mode s =0within the ÿnite time t r ,which satisÿes t r 6|s (0)|Á:(6)When the sliding mode s =0is reached,the system dy-namics is determined by the following nonlinear di erential equation:x 2+ÿx q=p 1=˙x 1+ÿx q=p1=0;(7)where x 1=0is the terminal attractor of the system (7).The ÿnite time t s that is taken to travel from x 1(t r )=0to x 1(t s +t r )=0is given byt s =−ÿ−1x 1(t r )d x 1x q=p 1=p ÿ(p −q )|x 1(t r )|1−q=p :(8)This means that,in the TSM manifold (7),both the system states x 1and x 2converge to zero in ÿnite time.It can be seen in the TSM control (5)that the secondterm containing x q=p −11x 2may cause a singularity to occur if x 2=0when x 1=0.This situation does not occur inthe ideal sliding mode because when s =0;x 2=−ÿx q=p1hence as long as q ¡p ¡2q ,i.e.1¡p=q ¡2,the term x q=p −11x 2is equivalent to x (2q −p )=p 1which is non-singular.The singularity problem may occur in the reaching phase when there is insu cient control to ensure that x 2=0while x 1=0.The TSM controller (5)cannot guarantee a bounded controlsignal for the case of x 2=0when x 1=0before the system states reach the TSM s =0.Furthermore,the singularity may also occur even after the sliding mode s =0is reached since,due to computation errors and uncertain factors,the system states cannot be guaranteed to always remain in the sliding mode especially near the equilibrium point (x 1=0;x 2=0),and the case of x 2=0while x 1=0may occur from time to time.This underlines the importance of addressing the singularity problem in conventional TSM systems.3.Non-singular terminal sliding mode controlIn order to overcome the singularity problem in the con-ventional TSM systems,several methods have been pro-posed.For example,one approach is to switch the sliding mode between TSM and linear hyperplane based sliding mode (Man &Yu,1997).Another approach is to transfer the trajectory to a pre-speciÿed open region where TSM control is not singular (Wu et al.,1998).These methods are adopting indirect approaches to avoid the singularity.In this paper,a simple NTSM is proposed,which is able to avoid this problem completely.The proposed NTSM model is de-scribed as follows:s =x 1+1ÿx p=q 2;(9)where ÿ;p and q have been deÿned in (2).One can easilysee that when s =0,the NTSM (9)is equivalent to (2)so that the time taken to reach the equilibrium point x 1=0when in the sliding mode is the same as in (8).Note that in using (9)the derivative of s along the system dynamics does not result in terms with negative (fractional)powers.This can be seen in the following theorem about the NTSM control.Theorem 1.For system (1)with the NTSM (9),if the control is designed asu =−b −1(x ) f (x )+ÿq px 2−p=q2+(l g +Á)sgn(s );(10)where 1¡p=q ¡2;Á¿0,then the NTSM manifold (9)will be reached in ÿnite time.Furthermore ,the states x 1and x 2will converge to zero in ÿnite time .Proof.For the NTSM (9),its derivative along the system dynamics (1)is ˙s =˙x 1+1ÿp q x p=q −12˙x 2=x 2+1ÿp q x p=q −12˙x 2=x 2+1ÿp q x p=q −12(f (x )+g (x )+b (x )u )Y.Feng et al./Automatica38(2002)2159–21672161=x2+1ÿpqx p=q−12g(x)−ÿqpx2−p=q2−(l g+Á)sgn(s)=1ÿpqx p=q−12(g(x)−(l g+Á)sgn(s))thens˙s=1ÿpqx p=q−12(g(x)s−(l g+Á)sgn(s)s)6−1ÿpqÁx p=q−12|s|:Since p and q are positive odd integers and1¡p=q¡2,there is x p=q−12¿0for x2=0.Let (x2)=(1=ÿ)(p=q)Áx p=q−12.Then it hass˙s6− (x2)|s|(x2)¿0for x2=0:(11)Therefore,for the case x2=0,the condition for Lya-punov stability is satisÿed.The system states can reach the sliding mode s=0withinÿnite ing the following ar-guments can easily prove this:substituting the control(10) into system(1)yields˙x2=−ÿqpx2−p=q2+g(x)−(l g+Á)sgn(s):Then,for x2=0,it is obtained˙x2=g(x)−(l g+Á)sgn(s):For both s¿0and s¡0,it is obtained˙x26−Áand ˙x2¿Á,respectively,showing that x2=0is not an attractor.It also means that there exists a vicinity of x2=0such that for a small ¿0such that|x2|¡ ,there are˙x26−Áfor s¿0 and˙x2¿Áfor s¡0,respectively.Therefore,the crossing of the trajectory from the boundary of the vicinity x2= to x2=− for s¿0,and from x2=− to x2= for s¡0occurs inÿnite time.For other regions where|x2|¿ ,it can be easily concluded from(11)that the switching line s=0can be reached inÿnite time since we have˙x26−Áfor s¿0 and˙x2¿Áfor s¡0.The phase plane plot of the system is shown in Fig.1.Therefore,it is concluded that the sliding mode s=0can be reached from anywhere in the phase plane inÿnite time.Once the switching line is reached,one can easily see that NTSM(9)is equivalent to the TSM(2),so the time taken to reach the equilibrium point x1=0in the sliding mode is the same as in(8).Therefore,the NTSM manifold(9)can be reached inÿnite time.The states in the sliding mode will reach zero inÿnite time.This completes the proof.Remark1.It should be noted that the NTSM control(10) is always non-singular in the state space since1¡p=q¡2.Remark2.In order to eliminate chattering,a saturation function sat can be used to replace the sign function sgn.The1Fig.1.The phase plot of the system.relationshipbetween the steady-state errors of the NTSM system and the width of the layer surrounding the NTSM manifold s(t)=0is given by(Feng,Han,Stonier,&Man, 2000;Feng,Yu,&Man,2001)|s(t)|6’⇒|x(t)|6’and|x(t)|6(2ÿ’)q=p for t→∞:(12)4.Non-singular terminal sliding mode control for rigid manipulatorsIn this section,a non-singular terminal sliding mode con-trol is designed for the rigid n-link robot manipulatorM(q) q+C(q;˙q)+g(q)= (t)+d(t);(13) where q(t)is the n×1vector of joint angular position,M(q) the n×n symmetric positive deÿnite inertia matrix,C(q;˙q) the n×1vector containing Coriolis and centrifugal forces, g(q)the n×1gravitational torque,and (t)n×1vector of applied joint torques that are actually the control inputs,and d(t)n×1bounded input disturbances vector.It is assumed that rigid robotic manipulators have uncertainties,i.e.:M(q)=M0(q)+ M(q);C(q;˙q)=C0(q;˙q)+ C(q;˙q);g(q)=g0(q)+ g(q);where M0(q);C0(q;˙q)and g0(q)are the estimated terms; M(q); C(q;˙q)and g(q)are uncertain terms.Then, the dynamic equation of the manipulator can be written in the following form:M0(q) q+C0(q;˙q)+g0(q)= (t)+ (t)(14)2162Y.Feng et al./Automatica 38(2002)2159–2167with(t )=− M (q ) q − C (q ;˙q )q − g (q ):(15)The following assumptions are made about the robot dy-namics: M (q ) ¡ 0;(16) C (q ;˙q ) ¡ÿ0+ÿ1 q +ÿ2 ˙q 2;(17) g (q ) ¡ 0+ 1 q ;(18) (t ) ¡ 0+ 1 q + 2 ˙q 2;(19) (t ) ¡b 0+b 1 q +b 2 ˙q 2;(20)where 0;ÿ0;ÿ1;ÿ2; 0; 1; 0; 1; 2;b 0;b 1;b 2are positivenumbers.Suppose that q r is the desired input of the robot mani-pulator and ˙q r is the derivative of q r .Deÿne ”(t )=q −q r ;˙”(t )=˙q −˙q r ;e (t )=[”T (t )˙”T (t )]T .Then,the error equation of the rigid robotic manipulator can be obtained as follows:˙e (t )=Ae +B ;(21)whereA = 0I 00 ;B =0I;=M −10(q )(−C 0(q ;˙q )−g 0(q )−M 0(q ) q r + (t )+ (t )):It can be observed that the error dynamics (21)is of form (13).The NTSM control strategy developed in Section 3can be applied.The result is summarized in the following theorem.Before proceeding further,the notation of the frac-tional power of vectors is introduced.For a variable vector z ∈R n ,the fractional power of vectors is deÿned asz q=p =(z q=p 1;z q=p 2;:::;z q=p n )T;˙z q=p =(˙z q=p 1;˙z q=p 2;:::;˙zq=p n )T:Theorem 2.For the rigid n -link manipulator (14),if the NTSM manifold is chosen as s =”+C 1˙”p=q ;(22)where C 1=diag [c 11;:::;c 1n ]is a design matrix ,and the NTSM control is designed as follows ,then the system tracking error ”(t )will converge to zero in ÿnite time . = 0+u 0+u 1;(23)where0=C 0(q ;˙q )+g 0(q )+M 0(q ) q r ;(24)u 0=−q pM 0(q )C −11˙”2−p=q;(25)u 1=−q p [s T C 1diag (˙”p=q −1)M −10(q )]T s T C 1diag (˙”p=q −1)M −10(q )×[ s C 1diag (˙”p=q −1)M −10(q ) (b 0+b 1 q+b 2 ˙q 2)];(26)where b 0;b 1;b 2are supposed to be known parameters as in (20).Proof.Consider the following Lyapunov functionV =12s Ts :Di erentiating V with respect to time,and substituting (23)–(26)into it yields˙V =s T ˙s =s T ˙”+p qC 1diag (˙”p=q −1) ”=s T ˙”+p q C 1diag (˙”p=q −1)M −10(q )(u 1(t )+u 0(t ))+ (t ))=s T p q C 1diag (˙”p=q −1)M −10(q )(u 1(t )+ (t )) =−p qs C 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2)+p qs T C 1diag (˙”p=q −1)M −10(q ) (t )6−p qs C 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2)+p qs C 1diag (˙”p=q −1)M −10(q ) (t ) =−p qC 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2− (t ) ) s that is˙V 6−Á(t ) s ¡0for s =0;(27)where Á(t )=p qC 1diag (˙”p=q −1)M −10(q ) ×{(b 0+b 1 q +b 2 q 2)− (t ) }¿0:Therefore,according to the Lyapunov stability criterion,the NTSM manifold s (t )in (22)converges to zero in ÿ-nite time.On the other hand,if s =”+C 1˙”p=q =0are reached as shown in Theorem 1,then the output trackingY.Feng et al./Automatica38(2002)2159–21672163 error of the robot manipulator”(t)=q−q r will convergeto zero inÿnite time.This completes the proof.Remark3.The NTSM control proposed in Theorem2solves the control of the rigid n-link manipulator,that repre-sents a special class of problems.The method proposed canbe extended to a class of n-order(n¿2)nonlinear dynam-ical systems,that represents a broader class of problems:˙x1=f1(x1;x2);˙x2=f2(x1;x2)+g(x1;x2)+B(x1;x2)u;(28)where x1=(x11;x12;:::;x1n)T∈R n;x2=(x21;x22;:::;x2n)T∈R n;f1and f2are smooth vector functions and g rep-resents the uncertainties and disturbances satisfyingg(x1;x2) 6l g where l g¿0;B is a non-singular ma-trix and u=(u1;u2;:::;u n)T∈R n is the control vector.It is further assumed that(x1;x2)=(0;0)if and only if(x1;˙x1)=(0;0).Note that many practical dynamical sys-tems satisfy this condition,for example,the mechanicalsystems.Robotic systems are certainly a special case of(28).Actually,the robotic system(14)is not in the form of(28),but it can be transformed to such form by the coordi-nates change.So,the proposed algorithm in the paper can beapplied to any plant,which can be transformed to(28).TheNTSM for system(28)can be designed as follows.Chooses=x1+ ˙x p=q1;(29)where =diag( 1;:::; n);( i¿0)for i=1;:::;n,and˙x p=q1is represented as˙x p=q1=(x p1=q111;:::;x p n=q n1n)T:If the NTSM control is designed as in(30),then the high-order nonlinear dynamical systems(28)will converge to the NTSM and the equilibrium point inÿnite time,re-spectively,u=−@f1@x2B(x1;x2)−1l g@f1@x2+Áss+@f1@x1f1(x1;x2)+@f1@x2f2(x1;x2)+ −1 −1diag(x2−p1=q q11;:::;x2−p n=q n1n);(30)where =diag(p1=q1;:::;p n=q n);p i and q i are positive odd integers and q i¡p i¡2q i for i=1;:::;n.5.Simulation studiesThe section presents two studies:one is the comparison study of performance between NTSM and TSM,and the other an application to a robot control problem.-0.0500.050.10.150.20.250.3-0.4-0.20.20.40.60.81.0x1x2Fig.2.Phase plot of NTSM system.parison studyIn order to analyze the e ectiveness of the NTSM control proposed and to compare NTSM with TSM,consider the simple second-order dynamical system below:˙x1=x2;˙x2=0:1sin20t+u:(31) The NTSM and TSM are chosen as follows:s NTSM=x1+x5=32;s TSM=x2+x3=51:Three control approaches are adopted:NTSM control, TSM control,and indirect NTSM control.The NTSM con-trol is designed according to(10)and NTSM(9),and TSM control is designed according to(5)and TSM(2).The in-direct NTSM control is designed in the same way as TSM, with only one di erence,that is when|x1|¡ ,let p=q, and is selected as0.001(Man&Yu,1997).Three sys-tems achieve the same terminal sliding mode behavior.So, only the phase plane response of the NTSM control system is provided,as shown in Fig.2.The control signals for the three kinds of systems are shown in Figs.3–5.It can be ob-viously seen some valuable facts.No singularity occurs at all in the case of NTSM control.When the trajectory crosses the x1=0axis,singularity occurs in the case of TSM con-trol.For the indirect NTSM control,although singularity is avoided by switching from the TSM to linear sliding mode, the e ect of the singularity can be seen,especially when decreases to zero.However when is relatively large, the sliding mode of the system is switching between TSM and the linear plane based sliding mode,and the advantage of TSM system is lost.Therefore,from the results of the above simulations,the occurrence of singularity problem in the TSM system,the drawback of the indirect NTSM,and the e ectiveness of the NTSM in avoiding singularity,are observed,respectively.2164Y.Feng et al./Automatica 38(2002)2159–21670.51.0 1.52.02.5-8-7-6-5-4-3-2-1012time (sec.)uFig.3.Control signal of NTSM system.0.51.0 1.52.02.5-90-80-70-60-50-40-30-20-10010time(sec.)uFig.4.Control signal of TSM system.5.2.Control of a robotA simulation with a two-link rigid robot manipulator (seeFig.6)is performed for the purpose of evaluating the perfor-mance of the proposed NTSM control scheme.The dynamic equation of the manipulator model in Fig.6is given by a 11(q 2)a 12(q 2)a 12(q 2)a 22q 1 q 2 +−ÿ12(q 2)˙q 21−2ÿ12(q 2)˙q 1˙q 2ÿ12(q 2)˙q 22+ 1(q 1;q 2)g 2(q 1;q 2)g =1 2;(32)0.51.0 1.52.02.5-8-7-6-5-4-3-2-1012time(sec.)uFig.5.Control signal of indirect TSMsystem.Fig.6.Two-link robot manipulator model.wherea 11(q 2)=(m 1+m 2)r 21+m 2r 22+2m 2r 1r 2cos(q 2)+J 1;a 12(q 2)=m 2r 22+m 2r 1r 2cos(q 2);a 22=m 2r 22+J 2;ÿ12(q 2)=m 2r 1r 2sin(q 2);1(q 1;q 2)=((m 1+m 2)r 1cos(q 2)+m 2r 2cos(q 1+q 2)); 2(q 1;q 2)=m 2r 2cos(q 1+q 2):The parameter values are r 1=1m ;r 2=0:8m ;J 1=5kg m ;J 2=5kg m ;m 1=0:5kg ;m 2=1:5kg.The desired reference signals are given by q r 1=1:25−(7=5)e −t +(7=20)e −4t ;q r 2=1:25+e −t −(1=4)e −4t :The initial values of the system are selected as q 1(0)=1:0;q 2(0)=1:5;˙q 1(0)=0:0;˙q 2(0)=0:0:Y.Feng et al./Automatica 38(2002)2159–216721650123456789100.20.40.60.81.01.21.41.6time(sec)O u t p u t t r a c k i n g o f j o i n t 1( r a d )Fig.7.Output tracking of joint 1using a boundary layer.123456789101.21.31.41.51.61.71.81.92.0time(sec)O u t p u t t r a c k i n g o f j o i n t 2( r a d )Fig.8.Output tracking of joint 2using a boundary layer.The nominal values of m 1and m 2are assumed to be ˆm 1=0:4kg ;ˆm 2=1:2kg :The boundary parameters of system uncertainties in (20)are assumed to be b 0=9:5;b 1=2:2;b 2=2:8:Suppose the tracking error and the 1st tracking error are tobe |˜q i |60:001and |˙˜q i |60:024;i =1,2,where ˜q i =q i −q riand ˙˜q i =˙q i −˙q ri ;i =1,ing the above performance index,it can be determined the parameters of NTSM manifolds.According to (12),it is obtained that |˜q i |6’i ;i =1;2:Let ’i =0:001;i =1;2(33)012345678910-15-10-5051015202530time(sec)C o n t r o l i n p u t o f j o i n t 1( N m )Fig.9.Control of joint 1using a boundary layer.12345678910-14-12-10-8-6-4024time(sec)C o n t r o l i n p u t o f j o i n t 2 (N m )Fig.10.Control of joint 2using a boundary layer.the tracking error of the system |˜q i |can be guaranteed.Onthe other hand,according to (12),it is obtained that |˙˜q i |6(2ÿ’i )q=p ;i =1;2:Let(2ÿ’i )q=p 60:024;i =1;2;thenq p6log 0:024log(2ÿ’i );i =1;2:(34)For simplicity,let ÿi =1;i =1;2.Then from (34),it is obtained thatq p 6log 0:024log(2×1×0:001)=0:60015;i =1;2:(35)2166Y.Feng et al./Automatica 38(2002)2159–2167-0.100.10.20.30.40.50.60.70.80.9-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1e1(t)(rad)d e 1/d t (r a d /s )Fig.11.Phase plot of tracking error of joint 1.-0.5-0.4-0.3-0.2-0.10.100.20.30.40.50.6e2(t)(rad)d e 2/d t (r a d /s )Fig.12.Phase plot of tracking error of joint 2.Let qp=0:6:Now,the parameters of the TSM can be obtained as:q =3;p =5(there are many other options as well).Finally,the NTSM models are obtained as follows:s 1=˜q 1+˙˜q 5=31=0;s 2=˜q 2+˙˜q 5=32=0:In order to eliminate the chattering,the boundary layermethod is adopted (Slotine &Li,1991)in the NTSM con-trol.The simulation results are shown in Figs.7–12.Figs.7and 8show the output tracking of joints 1and 2.Figs.9and 10depict the control signals of joints 1and 2,respec-tively.Figs.11and 12show the phase plot of tracking error of joints 1and 2,respectively.One can easily see that the system states track the desired reference signals.First,theoutput tracking errors of the system reach the terminal slid-ing mode manifold s =0in ÿnite time,then they converge to zero along s =0in ÿnite time.It can be clearly seen that neither singularity nor chattering occurs in the two control signals.6.ConclusionsIn this paper,a global non-singular TSM controller for a second-order nonlinear dynamic systems with parameter uncertainties and external disturbances has been proposed.The time taken to reach the manifold from any initial sys-tem states and the time taken to reach the equilibrium point in the sliding mode have been proved to be ÿnite.The new terminal sliding mode manifold proposed can enable the elimination of the singularity problem associated with con-ventional terminal sliding mode control.The global NSTM controller proposed has been used for the control design of an n -degree-of-freedom rigid manipulator.Simulation results are presented to validate the analysis.The proposed controller can be easily applied to practical control of robots as given the advances of microprocessors,the vari-ables with fractional power can be easily built into control algorithms.ReferencesBhat,S.P.,&Bernstein, D.S.(1997).Finite-time stability of homogeneous systems.Proceedings of American control conference (pp.2513–2514).Feng,Y.,Han,F.,Yu,X.,Stonier,D.,&Man,Z.(2000).Tracking precision analysis of terminal sliding mode control systems with saturation functions.In X.Yu,J.-X.Xu (Eds.),Advances in variable structure systems :Analysis,integration and applications (pp.325–334).Singapore:World Scientiÿc.Feng,Y.,Yu,X.,&Man,Z.(2001).Non singular terminal sliding mode control and its applications to robot manipulators.Proceedings of 2001IEEE international symposium on circuits and systems ,Vol.III (pp.545–548).Sydney,May 2001.Haimo,V.T.(1986).Finite time controllers.SIAM Journal of Control and Optimization ,24(4),760–770.Man,Z.,Paplinski,A.P.,&Wu,H.(1994).A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators.IEEE Transactions on Automatic Control ,39(12),2464–2469.Man,Z.,&Yu,X.(1997).Terminal sliding mode control of mimo linear systems.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications ,44(11),1065–1070.Slotine,J.E.,&Li,W.(1991).Applied non-linear control .Englewood Cli s,NJ:Prentice-Hall.Tang,Y.(1998).Terminal sliding mode control for rigid robots.Automatica ,34(1),51–56.Utkin,V.I.(1992).Sliding modes in control optimization .Berlin,Heidelberg:Springer.Wu,Y.,Yu,X.,&Man,Z.(1998).Terminal sliding mode control design for uncertain dynamic systems.Systems and Control Letters ,34,281–288.Yu,X.,&Man,Z.(1996).Model reference adaptive control systems with terminal sliding modes.International Journal of Control ,64(6),1165–1176.Yurl,B.S.,&James,M.B.(1988).Continuous sliding mode control.Proceedings of American Control Conference (pp.562–563).Y.Feng et al./Automatica 38(2002)2159–21672167Yong Feng received the B.S.degree from the Department of Control Engineering in 1982,and M.S.degree from the Depart-ment of Electrical Engineering in 1985and Ph.D.degree from the Department of Con-trol Engineering in 1991,in Harbin Insti-tute of Technology,China,respectively.He has been with the Department of Electri-cal Engineering,Harbin Institute of Tech-nology since 1985,and is currently a Pro-fessor.He was a visiting scholar in the Faculty of Informatics and Communication,Australia,from May 2000to November 2001.He has authored and co-authored over 50journal and conference papers.He has published 3books.He has completed over 10research projects,including process control,arc welding robot,climbing wall robot,CNC system,a direct drive motor and its control system,the electronics and simulation of CCD digital camera,and so on.His current research interests are nonlinear control systems,sampled data systems,robot control,digital camera modelling andsimulation.Xinghuo Yu received B.Sc.(EEE)and M.Sc.(EEE)from the University of Sci-ence and Technology of China in 1982and 1984respectively,and Ph.D.degree from South-East University,China in 1987.From 1987to 1989,he was Research Fellow with Institute of Automation,Chi-nese Academy of Sciences,Beijing,China.From 1989to 1991,he was a Postdoctoral Fellow with the Applied Mathematics De-partment,University of Adelaide,Australia.From 1991to 2002,he was with CentralQueensland University,Rockhampton,Australia where he was Lecturer,Senior Lecturer,Associate Professor then Professor of Intelligent Sys-tems and the Associate Dean (Research)of the Faculty of Informatics and Communication.Since March 2002,he has been with the School of Electrical and Computer Engineering at Royal Melbourne Institute of Technology,Australia,where he is a Professor,Director of Software and Networks,and Deputy Head of School.He has also held Visiting Profes-sor positions in City University of Hong Kong and Bogazici University(Turkey).He has recently been conferred as Honorary Professor of Cen-tral Queensland University.He is Guest Professor of Harbin Institute of Technology (China),Huazhong University of Science and Technology (China),and Southeast University (China).Professor Yu’s research inter-ests include sliding mode and nonlinear control,chaos and chaos control,soft computing and applications.He has published over 200refereed pa-pers in technical journals,books and conference proceedings.He has also coedited four research books “Complex Systems:Mechanism of Adapta-tion”(IOS Press,1994),“Advances in Variable Structure Systems:Anal-ysis,Integration and Applications”(World Scientiÿc,2001),“Variable Structure Systems:Towards the 21st Century”(Springer-Verlag,2002),“Transforming Regional Economies and Communities with Information Technology”(Greenwood,2002).Prof.Yu serves as an Associate Editor of IEEE Trans Circuits and Systems Part I and is on the Editorial Board of International Journal of Applied Mathematics and Computer Science.He was General Chair of the 6th IEEE International Workshopon Variable Structure Systems held in December 2000on the Gold Coast,Australia.He was the sole recipient of the 1995Central Queensland University Vice Chancellor’s Award forResearch.Zhihong Man received the B.E.degree from Shanghai Jiaotong University,China,the M.S.degree from the Chinese Academy of Sciences,and the Ph.D.from the Uni-versity of Melbourne,Australia,all in electrical and electronic engineering,in 1982,1986and 1993,respectively.From 1994to 1996,he was a Lecturer in the Department of Computer and Commu-nication Engineering,Edith Cowan Uni-versity,Australia.From 1996to 2000,he was a Lecturer and then a SeniorLecturer in the Department of Electrical Engineering,the University of Tasmania,Australia.In 2001,he was a Visiting Senior Fellow in the School of Computer Engineering,Nanyang Technological University,Singapore.Since 2002,he has been an Associate Professor of Computer Engineering at Nanyang Technological University.His research interests are in robotics,fuzzy logic control,neural networks,sliding mode control and adaptive signal processing.He has published more than 120journal and conference papers in these areas.。

Wheel Slip Control via Second-OrderSliding-Mode GenerationMatteo Amodeo,Antonella Ferrara,Senior Member,IEEE,Riccardo Terzaghi,and Claudio VecchioAbstract—During skid braking and spin acceleration,the driving force exerted by the tires is reduced considerably,and the vehicle cannot speed up or brake as desired.It may become very difficult to control the vehicle under these conditions.To solve this problem,a second-order sliding-mode traction controller is presented in this paper.The controller design is coupled with the design of a suitable sliding-mode observer to estimate the tire–road adhesion coefficient.The traction control is achieved by maintaining the wheel slip at a desired value.In particular, by controlling the wheel slip at the optimal value,the proposed traction control enables antiskid braking and antispin acceler-ation,thus improving safety in difficult weather conditions,as well as stability during high-performance driving.The choice of second-order sliding-mode control methodology is motivated by its robustness feature with respect to parameter uncertainties and disturbances,which are typical of the automotive context. Moreover,the proposed second-order sliding-mode controller,in contrast to conventional sliding-mode controllers,generates con-tinuous control actions,thus being particularly suitable for appli-cation to automotive systems.Index Terms—Chattering avoidance,higher order sliding modes,robust control,slip control,traction force control.N OMENCLATUREv x Longitudinal velocity(in meters per second).w f Front wheel angular velocity(in radians per second).w r Rear wheel angular velocity(in radians per second).T f Input torque on the front wheel(in newton meter).T r Input torque on the rear wheel(in newton meter).λf Front wheel slip ratio.λr Rear wheel slip ratio.F xf Longitudinal force at the front wheel(in newtons).F xr Longitudinal force at the rear wheel(in newtons).F zf Normal force on the front wheel(in newtons).F zr Normal force on the rear wheel(in newtons).F air Air drag force(in newtons).F roll Rolling resistance force(in newtons).m Vehicle mass(in kilograms).J f Front wheel moment of inertia(in kilograms per square meter).Manuscript received November8,2007;revised August4,2008and May19, 2009.First published November24,2009;current version published March3, 2010.The Associate Editor for this paper was A.Hegyi.M.Amodeo and R.Terzaghi are with Siemens S.p.a.,20128Milano,Italy (e-mail:matteo.amodeo@;riccardo.terzaghi@). A.Ferrara is with the Dipartimento di Informatica e Sistemistica,Universitàdegli studi di Pavia,27100Pavia,Italy(e-mail:antonella.ferrara@unipv.it). C.Vecchio is with the Dipartimento di Informatica e Sistemistica,Universitàdegli studi di Pavia,27100Pavia,Italy,and also with Temis s.r.l.,20011 Corbetta,Italy(e-mail:claudio.vecchio@;claudio.vecchio@ unipv.it).Digital Object Identifier10.1109/TITS.2009.2035438J r Rear wheel moment of inertia(in kilograms per square meter).R f Front wheel radius(in meters).R r Rear wheel radius(in meters).c x Longitudinal wind drag coefficient(in kilograms permeter).f roll Rolling resistance coefficient.l f Distance from the front axle to the center of gravity(in meters).l r Distance from the rear axle to the center of gravity(in meters).l h Height of the center of gravity(in meters).μp Road adhesion coefficient.ˆμp Estimated road adhesion coefficient.I.I NTRODUCTIONI N RECENT years,numerous different vehicle active controlsystems have been investigated and implemented in pro-duction[1].Among them,the traction control of vehicles is becoming increasingly important due to recent research efforts on intelligent transportation systems,particularly on automated highway systems,and on automated driver-assistance systems (see,for instance,[2]–[6]and the references therein).The objective of traction control systems is to prevent the degradation of vehicle performances,which occur during skid braking and spin acceleration.As a result,the vehicle perfor-mance and stability,particularly under adverse external condi-tions such as wet,snowy,or icy roads,are greatly improved. Moreover,the limitation of the slip between the road and the tire significantly reduces the wear of the tires.The traction force produced by a wheel is a function of the wheel slipλ,of the normal force acting on a wheel F z,and of the adhesion coefficientμp between road and tire,which,in turn,depends on road conditions[7],[8].Since the adhesion coefficientμp is unknown and time varying during driving,it is necessary to estimate such a parameter on the basis of the data acquired by the sensors.Because of its direct influence on the vehicle traction force,the wheel slipλis regarded as the controlled variable in the traction force control system.The design of such a control system is based on the assumption that the vehicle velocity and the wheel angular velocities are both available online by direct measurements.As the wheel angular velocity can easily be measured with sensors,only the vehicle velocity is needed to calculate the wheel slipλ. The vehicle longitudinal velocity can be directly measured[9], [10],indirectly measured[11],and/or estimated through the use of observers[12],[13].Since the problem of measuring1524-9050/$26.00©2009IEEEthe longitudinal velocity is out of the scope of this paper,we assume that both the vehicle velocity and the wheel angular velocities are directly measured.The traction control problem is addressed in this paper.The main difficulty arising in the design of a traction force control system is due to the high nonlinearity of the system and the presence of disturbances and parameter uncertainties[6],[14].A robust control methodology needs to be adopted to solve the problem in question.In this paper,we rely on sliding-mode control[15],[16]because of its appreciable properties,which make it particularly suitable to deal with uncertain nonlinear time-varying systems.Different sliding-mode controllers have been proposed in the literature to solve the problem of controlling the wheel slip.For instance,sliding-mode control is used to steer the wheel slip to the optimal value to produce the maximum braking force,and a sliding-mode observer for the longitudinal traction force is proposed in[6].A sliding-mode-based observer for the vehicle speed is proposed in[13].In[5],a sliding-mode control law that uses an online estimation of the tire–road adhesion coefficient is presented.Other different sliding-mode approaches to the traction control problem have been proposed(see,for instance, [17]–[21]and the reference therein).However,the conventional sliding-mode control generates a discontinuous control action that has the drawback of producing high-frequency chattering, with the consequent excessive mechanical wear and passen-gers’discomfort,due to the propagation of vibrations through-out the different subsystems of the controlled vehicle.To reduce the vibrations induced by the controller,a possible solution consists of the approximation of the discontinuous control signals with continuous signals.This is,for instance,the solution adopted in[5]and[14].However,this kind of solution only generates pseudosliding modes[15],[22].This means that the controlled system state evolves in the boundary layer of the ideal sliding subspace and features a dynamical behavior different from that attainable if ideal sliding modes could be generated.Therefore,even,if from a practical viewpoint,this solution can produce acceptable results,the robustness features with respect to matched uncertainties[22]are lost.The idea investigated in this paper to circumvent the incon-venience of the vibrations induced by sliding-mode controllers is to exploit the positive features of second-order sliding-mode control[23].Second-order sliding-mode controllers feature higher accuracy with respect tofirst-order sliding-mode control and generate continuous control actions,since the discontinuity is confined to the derivative of the control signal while keeping the robustness feature typical of conventional sliding-mode controllers[16].Nevertheless,the generated sliding modes are ideal,in contrast to what happens for solutions that rely on con-tinuous approximations of the discontinuous control laws[16]. The particular traction control problem addressed in this paper is the so–called fastest acceleration/deceleration control (FADC)problem.It can be formulated as the problem of maximizing the magnitude of the traction force to produce the maximum acceleration while driving and the smallest stopping distance during braking,even on a possibly slippery road. This is attained by regulating the wheel slip ratio at the value corresponding to the maximum/minimum traction force.Since the reference slip ratios depend on the adhesion coefficientμp, which is unknown and time varying during driving,the con-troller design is coupled with the design of a suitable sliding-mode observer to estimate the tire–road adhesion coefficient. This makes the performance of the proposed control system insensitive to possible variations of the road conditions,since such variations are compensated online by the controller.This paper is organized as follows:Section II introduces the model of the vehicle dynamics,specifies the assumptions, and states the control objectives.The proposed second-order sliding-mode slip controller is presented in Section III.A sliding-mode observer for the tire–road adhesion coefficient is proposed in Section IV.In Section V,the FADC problem is described.Simulation results relevant to the proposed controller are reported in Section VI,whereas somefinal comments are gathered in the last section.II.V EHICLE L ONGITUDINAL D YNAMICSIn this paper,a nonlinear model of the vehicle is adopted[7]. The vehicle is modeled as a rigid body,and only longitudinal motion is considered.The difference between the left and right tires is ignored,making reference to a so-called bicycle model. The lateral,yawing,pitch,and roll dynamics,as well as actuator dynamics,are also neglected.The resulting equations of motion for the vehicle arem˙v x=F xf(λf,F zf)+F xr(λr,F zr)−F loss(v x)(1)J f˙ωf=T f−R f F xf(λf,F zf)(2)J r˙ωr=T r−R r F xr(λr,F zr)(3)F loss(v x)=F air(v x)+F roll=c x v2x·sign(v x)+f roll mg(4)F zf=l r mg−l h m˙v xl f+l r(5)F zr=l f mg+l h m˙v xl f+l r(6)where v x is the longitudinal velocity of the vehicle center of gravity,ωf andωr are the angular velocity of the front and rear wheels,T f and T r are the front and rear input torque,λf and λr are the slip ratio at the front and rear wheel,F xf and F xr are the front and rear longitudinal tire–road contact forces,F zf and F zr are the normal force on the front and rear wheels,F air is the air drag resistance,and F roll is the rolling resistance(see Fig.1).The vehicle parameters are the following:m is the vehicle mass,c x is the longitudinal wind drag coefficient,f roll is the rolling resistance coefficient,J f and J r are the front and rear wheel moments of inertia,R f and R r are the front and rear wheel radius,l f is the distance from the front axle to the center of gravity,l r is the distance from the rear axle to the center of gravity,and l h is the height of the center of gravity(see Fig.1). The normal force calculation method adopted in this paper[see (5)and(6)]is based on a static force model,as described in [8],ignoring the influence of suspension.This method gives a fairly accurate estimate of the normal force,particularly when the road surface is fairly paved and not bumpy.Fig.1.Vehicle model.The longitudinal slip λi ,i ∈{f,r }for a wheel is defined as the relative difference between a driven wheel angular velocity and the vehicle absolute velocity,i.e.,λi =ωi R i −v xωi R i,ωi R i >v x ,ωi =0,acceleration ωi R i −v xv x,ωi R i <v x ,v x =0,braking i ∈{f,r }.(7)The wheel slip dynamics during acceleration can be obtainedby differentiating (7)with respect to time,thus obtaining˙λi =f a i +h a iT i ,i ∈{f,r }(8)wheref a i =−˙v x R i ωi −v x F xiJ i ω2i,i ∈{f,r }(9)h a i =v xJ i R i ω2i,i ∈{f,r }.(10)The dynamics during braking can analogously be obtained bydifferentiating (7)for the brake situation and results in˙λi =f b i +h b iT i ,i ∈{f,r }(11)wheref bi =−R i ωi ˙v x v 2x −R 2iF xi J i v x,i ∈{f,r }(12)h b i =R ii x 2i,i ∈{f,r }.(13)The traction force F xi in the longitudinal direction generatedat each tire is a nonlinear function of the longitudinal slip λi ,of the normal force applied at the tire F zi ,and of the road adhesion coefficient μp [7].Different longitudinal tire–road friction models for vehicle motion control have been proposed in the literature (see [24]).In this paper,the so–called “Magic Formula”tire model developed by Bakker et al.[25]is con-sidered.This model is generally accepted as the most useful and viable model in describing the relationship between the slip ratio and the tire force.The model for the longitudinal force is as follows:F xi =f t (μp ,λi ,F zi ),i ∈{f,r }(14)III.S LIP C ONTROL D ESIGNAs previously mentioned,due to the high nonlinearity of the system and to the presence of time–varying parameters and uncertainties,typical of the automotive context,the control system is designed by relying on a robust control approach, i.e.,second-order sliding-mode control.The main advantage of second-order sliding-mode control[23]with respect to the first-order case[15]is that it features higher accuracy[16]and generates continuous control actions while keeping the same robustness properties with respect to matched uncertainties[22] and a comparable design complexity.As previously discussed,the controlled variable in the pro-posed traction force control system is the slip ratio at a wheel λi,i∈{f,r},because of its strong influence on the traction force.Indeed,it is possible to adjust the traction force produced by a tire F xi,i∈{f,r}to the desired value by controlling the wheel slip.Thus,the control objective of the control sys-tem is to make the actual slip ratioλi track the desired slip ratioλd,i.The sliding variables are chosen as the error between the current slip and the desired slip ratio,i.e.,s i=λei=λi−λd,i,i∈{f,r}.(17) As a consequence,the chosen sliding manifolds are given bys i=λei=λi−λd,i=0,i∈{f,r}(18) and the objective of the control is to design continuous control laws T i,i∈{f,r}that is capable of enforcing sliding modes on the sliding manifolds[see(18)]infinite time.Note that, once the sliding mode is enforced,the actual slip ratio correctlytracks the desired slip ratio since on the sliding manifoldλei =0,and the control objective is attained infinite time.Thefirst and second derivatives of the sliding variable s i in the acceleration case are given by˙s i=f a i+h a i T i−˙λd i,i∈{f,r}¨s i=ϕa i+h a i˙T i,i∈{f,r}(19) whereϕa i andγa i,i∈{f,r}are defined asϕa i=−¨v xR iωi+2˙v x˙ωiR iω2i−2v x˙ω2iR iω3i−¨λdi−v x˙FxiJ iω2i.(20)Note that the quantities h a i,i∈{f,r}are known.From(1)and(15),we get|˙v x|≤2Ψ−F loss(v x)m=f1(v x).(21)Taking into account thefirst time derivative of(1),(16),and (21),one has that|¨v x|≤2Γ−2|˙v x ||v x|m ≤2Γ−2f1(v x)|v x|m=f2(v x).(22)From(2),(3),and(15),it results in|˙ωi|≤Ψ−T iJ i =f3i(T i),i∈{f,r}.(23)Relying on(21)–(23),one has that the quantitiesϕa i,i∈{f,r}are bounded.From a physical viewpoint,this means that,whena constant torque T i,i∈{f,r}is applied,the second timederivative of the slip ratios is bounded.To apply a second-order sliding-mode controller,it is notnecessary for a precise evaluation ofϕa i to be available.In thesequel of this paper,it will only be assumed that suitable boundsΦa i(v x,ωi,T i)ofϕa i,i.e.,|ϕa i|≤Φa i(v x,ωi,T i),i∈{f,r}(24)are known.As for the braking case,the functionsϕb i andγb i can beobtained by following the same procedure previously describedfor the acceleration case.As forϕa i,ϕb i can be regarded asunknown bounded functions with known boundsΦb i(v x,ωi,T i),i.e.,|ϕb i|≤Φb i(v x,ωi,T i),i∈{f,r}.(25)To design a second-order sliding-mode control law,introducethe auxiliary variables y1,i=s i and y2,i=˙s i.Then,system(19)can be rewritten as˙y1,i=y2,i˙y2,i=ϕji+h ji˙Ti,i∈{f,r},j∈{a,b}(26)where˙T i can be regarded as the auxiliary control input[23].Theorem1:Given system(26),whereϕjisatisfies(24)and(25),and y2,i is not measurable,the auxiliary control law is˙Ti=−V i signs i−12s iM,i∈{f,r}(27)where the control gain V i is chosen such thatV i>2Φa i(v x,ωi,T i)/h a i,acceleration case2Φb i(v x,ωi,T i)/h b i,braking casei∈{f,r}(28)and s iM is a piecewise constant function representing the valueof the last singular point of s i(t)[i.e.,s iM is the value of themost recent maximum or minimum of s i(t)]that causes theconvergence of the system trajectory on the sliding manifolds i=˙s i=0infinite time.Proof:The control law[see(27)]is a suboptimal second-order sliding-mode control law.Therefore,by following a the-oretical development as that provided in[26]for the generalcase,it can be proved that the trajectories on the s i O˙s i plane areconfined within limit parabolic arcs,including the origin.Theabsolute values of the coordinates of the trajectory intersectionswith the s i-and˙s i-axes decrease in time.As shown in[26],under condition(28),the following relationships hold:|s i|≤|s iM|,|˙s i|≤|s iM|and the convergence of s iM(t)to zero takes place infinitetime[26].As a consequence,the origin of the plane,i.e.,s i=˙s i=0,is reached infinite since s i and˙s i are both boundedby max(|s iM|,|s iM|).This,in turn,implies that the sliperrorsλei ,i∈{f,r}are steered to zero as required to attainthe objective of the traction control problem.IV.T IRE–R OAD A DHESION C OEFFICIENT E STIMATE To identify theλ−F x curve corresponding to the actual road condition,the tire–road adhesion coefficientμp needs to be estimated.Different estimation techniques for this parameter have been proposed in the literature,and most of them are based on the Bakker–Pacejka Magic Formula model.For instance,in [27],a procedure for the real-time estimation ofμp is presented, whereas in[20],a scheme to identify different classes of roads with a Kalmanfilter and a least-square algorithm is presented. In[5]and[28],a recursive least-square algorithm[29]is adopted to estimate the tire–road adhesion coefficient.A dif-ferent approach is proposed in[30],where an extended Kalman filter is used to estimate the forces produced by the tires.A sliding-mode observer to estimate the longitudinal stiffness for a simplified linear tire–road interaction model was proposed in[6]and[31],while a dynamical tire–road interaction model with a nonlinear observer to estimate the adhesion coefficient has been proposed in[32].In this section,afirst-order sliding-mode observer for the online estimation of the adhesion coefficientμp is designed. The sliding-mode methodology has also been adopted to design the observer since it is applicable to nonlinear systems and has good robustness properties against disturbances,modeling inaccuracy,and parameter uncertainties[15].Following the approach proposed in[5],a simplified tire model is considered instead of(14),i.e.,F xi=μp f t(λi,F zi),i∈{f,r}.(29)To design the sliding-mode observer forμp relying on the so-called equivalent control method[22],introduce the sliding variablesμ=v x−ˆv x(30) whereˆv x is an estimate of the longitudinal velocity v x.The dynamics ofˆv x is chosen as˙ˆv x =1m(Ω−F loss(v x))(31)whereΩ=K sign(sμ)(32) is the control signal of the sliding-mode observer.In the sequel,for notation simplicity,the dependence of the tire force F x on the slip ratioλand the normal force F z has been omitted.By differentiating(30)and substituting(1),one has that˙sμ=˙v x−˙ˆv x=1m(F xf+F xr−K sign(sμ)).(33)From(14),the following relationship holds:F xf+F xr≤F zf+F zr=mg.(34)Relying on(33)and(34),if the gain K in(32)is chosen such thatK>mg≥F xf+F xr(35) then the so-called reaching condition[22]sμ˙sμ≤−η|sμ|,η∈I R+(36) is satisfied,and a sliding mode on the sliding manifold sμ=0 is attained infinite time.The tire–road adhesion coefficientμp can be estimated by taking into account the so-called equivalent controlΩeq,which is defined as the continuous control signal that maintains the system on the sliding surface sμ=0[15].The equivalent control can be calculated by setting the time derivative of the sliding variable˙sμequal to zero,i.e.,˙sμ=1m(F xf+F xr−Ωeq)=0(37) thus the equivalent controlΩeq is given byΩeq=F xf+F xr.(38) If we assume that the front and rear wheels are on the same road surface,which is true for many driving situations,then(38)can be rewritten asΩeq=F xf+F xr=μpf tf(λf,F zf)+f tr(λr,F zr).(39) The equivalent controlΩeq is close to the slow component of the real control and can be obtained byfiltering out the high-frequency component ofΩusing a low-passfilter[15],[22], that isτ˙ˆΩ+ˆΩ=Ω(40)Ωeq≈ˆΩ(41) whereτis thefilter time constant.Thefilter time constant should be chosen sufficiently small to preserve the slow compo-nents of the controlΩundistorted but large enough to eliminate the high-frequency component.From(39)and(41),the estimated tire–road adhesion coeffi-cientˆμp can be calculated asˆμp=ˆΩf tf(λf,F zf)+f tr(λr,F zr).(42) Note that,from(38)and(41),one has thatˆΩ=Fxf+F xr.(43) Thus,ˆΩcan also be regarded as a sliding-mode observer to estimate the total longitudinal force exerted by the vehicle.V.F ASTEST A CCELERATION/D ECELERATIONC ONTROL P ROBLEMThe particular traction-control problem taken into account in this paper is the so-called FADC problem.It can be formulated as the problem of maximizing the magnitude of the tractionforce to produce the maximum acceleration while driving and the smallest stopping distance during braking,even on a possi-bly slippery road.Looking at theλ−F x curve in Fig.2,the maximum ac-celeration can be attained by steering the slipλto the value corresponding to the positive peak of the curve,namely,λMax, i.e.,considering the i th axleλd,i=λMaxi.(44) Beyond this value,the wheels begin to spin,the longitudinal force produced decreases,and the vehicle cannot accelerate as desired.By maximizing the traction force between the tire and the road,the traction controller prevents the wheels from slipping and,at the same time,improves the vehicle’s stability and steerability.Similarly,the target slip to obtain the maximum braking force,i.e.,the minimum braking distance,is determined as the slip value corresponding to the minimum of theλ−F x curve,namely,λMin.Thus,the maximum braking force can be attained by the steering the tire slipλtoλMin,i.e.,considering the i th axleλd,i=λMini.(45)The position ofλMaxi varies,depending on the actualλi−F xicurve considered,and its value is generally unknown duringdriving.The same holds forλMini .As a consequence,the con-trol task has to include the online searching of the peak slip.In the proposed approach,this task is accomplished in two steps.1)The tire–road adhesion coefficientμp is estimated asdescribed in Section IV,and the currentλi−F xi curve is identified.2)For the acceleration case,the desired slip,i.e.,the slipratio corresponding to the maximum of the curve,is calculated by maximizing the functionˆF xi=f ti(ˆμp,λi,F zi)asλd,i=arg minλi −ˆF xi=arg minλi−f ti(ˆμp,λi,F zi).(46)As for the braking case,the desired slip ratio correspond-ing toλMini is calculated by minimizing the functionˆF xi,that isλd,i=arg minλif ti(ˆμp,λi,F zi).(47)Note that the minimum(maximum)of the functionˆF xi can be calculated,for instance,with a minimization algorithm without derivatives[34].Note that different strategies have been proposed in the litera-ture tofind the slip ratio corresponding to the maximum of the λ−F x curve(see,for instance,[3],[5],[6],and[35]).VI.S IMULATION R ESULTSThe traction control presented in this paper has been tested in simulation,considering a scenario with different road con-ditions.The vehicle is travelling at an initial velocity v x(0)= 20m/s,with initial slip ratiosλf(0)=λr(0)=0.02,and theTABLE IS IMULATION PARAMETERSh j i−ηi sign(s i)−f j i+˙λdii∈{f,r},j∈{a,b}(48)whereηi>0.As can be seen,in contrast with the proposed second-order sliding-mode controller,conventional sliding-mode con-trol laws produce discontinuous control inputs that generate high-frequency chattering,with the consequent excessive me-chanical wear and passengers’discomfort.To exploit the robustness feature of the proposed control scheme,the controlled system is tested in simulation in the presence of model uncertainties and disturbances and is com-pared with afirst-order sliding-mode solution,where the sgn(·) function is approximated with the sat(·)function,as in[5]. The nominal model parameters are as in Table I,whereas the real values for the mass,the wheel moment of inertia, and the wheel radius are m=1702kg,J f=J r=1.8kg m2, and R f=R r=0.5m,respectively.Moreover,to model some matched disturbances,the real control input is calculated as T i(t)=¯T i(t)+A sin(t),i∈{f,r}(49) where¯T i is the nominal control input given by(27),and A is the amplitude of the disturbances acting on the control input. Figs.11and12show the simulation results obtained with the proposed second-order sliding-mode control scheme with A= 300in(49).As expected,the proposed control scheme is robust against parameter uncertainties and matched disturbances.One can note that the tire–road adhesion coefficient iscorrectlyTABLE IIP ERFORMANCE I NDEXES[32]C.Canudas-De-Wit and R.Horowitz,“Observers for tire/road contactfriction using only wheel angular velocity information,”in Proc.38th Conf.Decision Control,Phoenix,AZ,1999,pp.3932–3937.[33]R.Marino and P.Tomei,“Global adaptive observer for nonlinear systemsviafiltered transformations,”IEEE Trans.Autom.Control,vol.37,no.8, pp.1239–1245,Aug.1992.[34]R.P.Brent,Algorithms for Minimization Without Derivatives.Englewood Cliffs,NJ:Prentice-Hall,1973.[35]D.Hong,P.Yoon,H.Kang,I.Hwang,and K.Huh,“Wheel slip controlsystems utilizing the estimated tire force,”in Proc.Amer.Control Conf.,Minneapolis,MN,2006,pp.5873–5878.Matteo Amodeo was born in Vizzolo Predabissi, Italy.He received the Master’s degree in computer engineering from the University of Pavia,Pavia, Italy,in2006.Since January2007,he has been with Siemens S.p.a.,Sector BT FSP-DMS,Milano,Italy.His re-search activities are mainly in the area of sliding-mode control applied to automotivecontrol.Antonella Ferrara(S’86–M’88–SM’03)was bornin Genova,Italy.She received the Laurea degreein electronic engineering and the Ph.D.degree incomputer science and electronics from the Universityof Genova in1987and1992,respectively.In1992,she was an Assistant Professor withthe Department of Communication,Computer andSystem Sciences,University of Genova.In1998,she was an Associate Professor of automatic controlwith the Universitàdegli studi di Pavia,Pavia,Italy.Since January2005,she has been a Full Professor of automatic control with the Department of Computer Engineering and Systems Science,Universitàdegli studi di Pavia.She has authored or coauthored more than230papers,including more than70international journal papers. Her research activities are mainly in the area of sliding-mode control with application to automotive control,process control,and robotics.Dr.Ferrara is a Senior Member of the IEEE Control Systems Society and a member of the IEEE Technical Committee on Variable Structure and Sliding-Mode Control,the IEEE Robotics and Automation’s Technical Committee on Autonomous Ground Vehicles and Intelligent Transportation Systems,and the IFAC Technical Committee on Transportation Systems.From2000to2004, she was an Associate Editor of the IEEE T RANSACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY.At present,she is an Associate Editor of the IEEE T RANSACTIONS ON A UTOMATIC C ONTROL.She has been a member of the International Program Committee of numerous international conferences and events.As a student at the Faculty of Engineering,University of Genova,she received the“IEEE North Italy Section Electrical Engineering Student Award”in1986.Riccardo Terzaghi was born in Vizzolo Predabissi, Italy.He received the Master’s degree in computer engineering from the University of Pavia,Pavia, Italy,in2006.Since January2007,he has been with Siemens S.p.a.,Sector BT FSP-DMS,Milano,Italy.His re-search activities are mainly in the area of higher order sliding-mode control and robust control with application to automotivesystems.Claudio Vecchio received the Master’s degree in computer engineering and the Ph.D.degree from the Universitàdegli studi di Pavia,Pavia,Italy,in2005 and2008,respectively.Since November2008,he has been with Temis s.r.l.,Corbetta,Italy.He is also with the Dipartimento di Informatica e Sistemistica,Universitàdegli studi di Pavia.His research interests are mainly in the area of higher order sliding-mode control and robust and nonlinear control,with application to automotive control.。

感应电机二阶滑模次优算法定子磁链观测器设计潘月斗;陈泽平;郭映维【摘要】提出了基于二阶滑模次优算法的感应电机定子磁链观测方法,设计了定子磁链观测器,并应用到感应电机直接转矩控制中.本文设计的磁链观测器,通过准确的跟踪电流及其变化率,从而实现对转子磁链的准确估算,然后利用转子磁链与定子磁链的关系,估算出定子磁链.由于本文设计的定子磁链观测器是一个多输入多输出(MIMO)系统,稳定性分析非常复杂,为此将磁链估算误差的微分看作扰动处理,从而将MIMO的观测器模型分解成两个独立的单输入单输出(SISO)系统,简化了稳定性分析.将该观测器用于感应电机直接转矩控制中,达到了很好的控制效果.仿真和实验验证了该方法的有效性.【期刊名称】《控制理论与应用》【年(卷),期】2015(032)005【总页数】5页(P641-645)【关键词】感应电机;二阶滑模;次优算法;电流观测;磁链观测;直接转矩控制【作者】潘月斗;陈泽平;郭映维【作者单位】北京科技大学自动化学院,北京100083;北京科技大学钢铁流程先进控制教育部重点实验室,北京100083【正文语种】中文【中图分类】TM343感应电机被广泛应用于工农业生产、国防、科技及社会生活等各个方面,随着直接转矩控制和矢量控制技术的出现,使其逐渐进入了伺服控制领域[1].相对于矢量控制,直接转矩控制方法直接把转矩作为被控量,并由电流和定子磁链估算,无需进行磁场定向和矢量变换,更为简单和实用,具有快速的动态响应能力[2].直接转矩控制中,定子磁链观测值的精确度直接影响控制效果[3].定子磁链观测的基本方法有电压模型法和电流模型法.电压模型法结构简单,观测时仅需确定定子电阻.但是电压模型法在运算过程中需开环积分(纯积分),微小的直流偏移误差和初始值误差都将导致积分饱和[4].电流模型法可解决电压模型积分漂移和无法建立初始磁链的问题,但观测精度与转速相关,易受电动机转速变化的影响[5].为了更好的观测磁链,已提出了很多方法,如滑模变结构方法[6–7]、自适应方法[8]、卡尔曼滤波器方法[9–10]、神经网络方法[11]等.相比其他方法,滑模变结构方法对系统的不确定性因素具有较强的鲁棒性和抗干扰性,同时控制设计简单,物理上易于实现,因此得到广泛应用.但是在实际应用中,滑模变结构控制也存在一些问题,其中最主要的是抖振现象[12].近些年提出的高阶滑模控制理论[13],是对传统滑模控制理论的进一步推广.相比传统滑模,高阶滑模不仅保持了传统滑模的优点,同时抑制了系统的抖振,除去了相对阶的限制,并且提高了控制精度.二阶滑模控制是目前应用最广泛的高阶滑模控制方法,因为它的控制器结构简单且所需要的信息不多.二阶滑模控制中常见的4种算法有:twisting(螺旋)算法、sub-optimal(次优)算法、prescribed convergence law(给定收敛律)算法和Super-Twisting(超螺旋)算法.本文设计了一种基于二阶滑模次优算法的感应电机定子磁链观测器.将磁链估算误差的微分看作扰动处理,从而将MIMO的观测器模型分解成两个独立的SISO系统,简化了稳定性分析.将该观测器用于感应电机直接转矩控制中,达到了很好的控制效果.仿真及实验结果验证了该方法的有效性.设感应电机的磁路是线性的,忽略铁损的影响,在静止坐标系(α–β)下,感应电机的数学模型的状态方程为[14]δ=ηRs+Lmλθ;isα,isβ,usα,usβ,ψrα,ψrβ分别为α轴和β轴的定子电流、定子电压和转子磁链;ωr为转子电角速度;Ls,Lr,Lm分别为定子电感、转子电感和定转子间互感;Rs,Rr分别为定子电阻和转子电阻.定子磁链和转子磁链存在如下关系[15]:设计如下感应电机转子磁链观测器:其中:分别为定子电流和转子磁链的状态估计变量,vα和vβ为控制信号,分别为α轴和β轴的定子电流观测误差.定子电压和定子电流usα,usβ,isα,isβ都是可以检测到的,定子电压是原实际系统(感应电机)的输入量,定子电流可作为原实际系统的输出量;针对此观测器而言,定子电流检测量isα,isβ作为给定输入量(也作为干扰输入的一部分),定子电压检测量usα,usβ以及转子电角速度看作干扰输入的一部分;,作为观测器的反馈量.式(1)减式(2),可以得到定子电流和转子磁链观测误差方程电流观测误差方程写成如下形式:由式(5)可知,电流误差方程系统相对于控制信号v是1阶系统,因此可以采用二阶滑模控制,设计控制信号v,使得滑模变量s趋于零,并保持二阶滑动模态,即s==0.如果选取s=,采用二阶滑模控制,即可使得=0.二阶滑模次优算法(sub-optimal)形式如下:其中:s∗是最近的时间内,=0时s的值;k1,k2为控制参数,令s(t,x)=0为所定义的滑模面,控制目标是使系统的状态在有限时间内收敛到滑模流形s== 0.选取滑模面s=设计如下控制律:其中:对于式(5),将看作扰动处理,可将其分成α轴和β轴方向两个独立的SISO(单入单出)系统,如下:文献[16]给出了次优算法有限时间收敛的充分条件:其中Km,KM,C满足如下条件:对于本文设计的观测器系统,α轴方向分析如下:上式对时间求导,可得系统有限时间收敛的充分条件[16]如下:如果参数kα1,kα2满足式(9),则系统必能在有限时间内到达滑模面满足如下条件: β轴方向的稳定性分析同上.利用转子磁链观测器估算得到的转子磁链和定子电流,可估算定子磁链基于二阶滑模次优算法的感应电机定子磁链观测器系统框图如图1所示.为了检验所设计的基于二阶滑模次优算法的感应电机定子磁链观测器的有效性,进行了MATLAB仿真与实验.电机参数为:额定电压UN=220V,定子电阻Rs=94Ω,转子电阻Rr=83.9Ω,定子自感Ls= 5.387H,转子自感Lr=5.387H,互感Lm=5.082H,转动惯量J=0.105kg·m2.观测器控制参数为:kα1=kβ1=10,kα2=kβ2=5.电机施加220V,15Hz的三相交流电,在开环下空载运转,4s时,施加3N·m负载转矩.仿真时间7s,仿真结果如图2–5所示.从图3和图4可以看出,观测电流误差及其微分(由于实际对磁链观测误差有影响的是,所以图4实际是δ的值),在一定时间内渐近趋于0,从而说明了给二阶滑模次优算法控制的有效性.从图5可以看出,观测磁链在一定时间内达到稳定.为了验证基于二阶滑模次优算法的感应电机定子磁链观测器的有效性,将其应用到感应电机直接转矩控制中.电机参数与开环时一样,定子磁链给定值ψ=1Wb,给定转速600r/min.转速调节器采用PID控制,其中比例系数KP=10,积分系数KI= 0.001,微分系数KD=0.5.仿真时间20s,仿真结果如图6所示.为了验证二阶滑模次优算法定子磁链观测器的实际可行性,利用“电力电子与电气传动综合实验台”进行实验.实验台组成包括:功率挂箱、主控挂箱、加载控制箱、电动机、上位机,如图7所示.实验电机为鼠笼式三相异步电动机,参数与仿真时所用电机参数相同.转速给定值600r/min,实验结果如图8所示.从仿真和实验结果可以看出,二阶滑模次算法定子磁链观测器能够很好的观测定子磁链,电机转速也最终稳定在了给定值600r/min,从而证明了本文所提出的基于二阶滑模次算法的感应电机定子磁链观测器的实际可行性.本文提出的二阶滑模次优算法定子磁链观测器,首次将二阶滑模次优算法应用到感应电机定子磁链观测器设计中,并将此观测器应用到直接转矩控制中.从仿真和实验结果可以看出,该观测器能够准确的估算定子磁链,将其用于感应电机直接转矩控制中,也达到了很好的控制效果.仿真实验验证了该方法的有效性.潘月斗(1966–),男,博士,副教授,目前研究方向为交流电动机智能控制理论研究及高速高精交流电动机驱动系统的计算机数字控制系统设计,E-mail:****************;陈泽平(1989–),男,硕士研究生,目前研究方向为电气传动及自动化,E-mail:**********************;郭映维(1990–),男,硕士研究生,目前研究方向为异步电机控制理论及数字化设计,E-mail:*****************.【相关文献】[1]PELLEGRINO G,GUGLIELMI P,ARMANDO E,et al.Selfcommissioning algorithm for inverter nonlinearity compensation in sensorless induction motor drives[J].IEEE Transactions on Industry Applications,2010,46(4):1416–1424.[2]张细政,王耀南,袁小芳,等.基于滑模与自适应观测器的感应电机非线性控制新策略[J].控制理论与应用,2010,27(6):753–760.(ZHANG Xizheng,WANG Yaonan,YUAN Xiaofang,et al.New nonlinear controller forinduction motor based on sliding-mode control and adaptive observer[J].Control Theory&Applications,2010, 27(6):753–760.)[3]张猛,肖曦,李永东.基于扩展卡尔曼滤波器的永磁同步电机转速和磁链观测器[J].中国电机工程学报,2007,27(36):36–40.(ZHANG Meng,XIAO Xi,LI Yongdong.Speed and flux linkage observer for permanent magnet synchronous motor based on EKF[J]. 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第40卷第11期2023年11月控制理论与应用Control Theory&ApplicationsV ol.40No.11Nov.2023基于有限时间扰动观测器的水厂加矾系统二阶滑模控制王冬生†,张鹏,孙锦昊,郭若寒,蒋国平(南京邮电大学自动化学院人工智能学院,江苏南京210023)摘要:水厂絮凝沉淀过程具有强非线性、不确定性和参数时变等特点,并且原水水质和水量突变等扰动容易对絮凝沉淀过程造成不利影响.本文提出了一种基于有限时间扰动观测器的加矾系统二阶滑模控制设计方法.首先,文章采用带有非光滑项的二阶滑模控制方法设计加矾系统反馈控制;然后,文章设计有限时间扰动观测器对原水水质和水量突变等扰动,以及絮凝沉淀过程强非线性、不确定性和参数时变等导致的模型不匹配进行估计,估计结果作为前馈补偿与反馈控制相结合;最后,理论分析证明了基于有限时间扰动观测器的二阶滑模控制方法的稳定性.仿真结果表明,本文所提出的复合控制方法有效提升了加矾系统的鲁棒性和抗扰动性能.关键词:加矾控制;有限时间扰动观测器;二阶滑模控制;抗扰动引用格式:王冬生,张鹏,孙锦昊,等.基于有限时间扰动观测器的水厂加矾系统二阶滑模控制.控制理论与应用, 2023,40(11):1965–1971DOI:10.7641/CTA.2022.20462Second-order sliding mode control based onfinite-time disturbance observer for alum dosing system of water plantWANG Dong-sheng†,ZHANG Peng,SUN Jin-hao,GUO Ruo-han,JIANG Guo-ping(School of Automation,School of Artificial Intelligence,Nanjing University of Posts and Telecommunications,Nanjing Jiangsu210023,China) Abstract:Theflocculation and sedimentation process of water plant has the characteristics of strong nonlinearity, uncertainty and time-varying parameters,and disturbances of sudden changes in raw water quality and waterflow are easy to adversely affect theflocculation and sedimentation process.This paper proposes a control design method of second-order sliding mode based on thefinite-time disturbance observer for alum dosing system.First,the feedback control of alum dosing system is designed by second-order sliding mode control method with non-smooth terms.Then,afinite-time disturbance observer is designed to estimate disturbances of sudden changes in raw water quality and waterflow,as well as model mismatch caused by strong nonlinearity,uncertainty,and time-varying parameters in theflocculation and sedimentation process.The estimation result is combined with feedback control as feedforward compensation.Finally,the theoretical analysis proves the stability of second-order sliding mode control method based on thefinite-time disturbance observer.The simulation results show that the composite control method proposed in this paper effectively improves the robustness and anti-disturbance performance of alum dosing system.Key words:alum dosing control;finite-time disturbance observer;second-order sliding mode control;anti-disturbance Citation:WANG Dongsheng,ZHANG Peng,SUN Jinhao,et al.Second-order sliding mode control based onfinite-time disturbance observer for alum dosing system of water plant.Control Theory&Applications,2023,40(11):1965–19711引言絮凝沉淀过程是水厂水质净化的重要环节,与出厂水水质安全密切相关.絮凝沉淀过程通过向原水中投加矾等絮凝剂去除原水中的悬浮杂质、胶体颗粒及附着于胶体颗粒上的细菌、病毒等有害物质.依据美国联邦环保局饮用水病毒去除技术标准,当滤后水浊度低于0.3NTU时,病毒去除率高达99%[1].加强对水厂加矾系统的有效控制,严格限制沉淀池出水浊度,有利于出厂水水质稳定和实现高品质饮用水目标.专家学者们对加矾系统控制问题进行了大量的研究和实践,提出了各种控制算法.流动电流法[2]和透光率脉动法[3]通过流动电流值和透光率检测跟踪絮凝收稿日期:2022−05−29;录用日期:2022−12−22.†通信作者.E-mail:***********************.cn;Tel.:+86189****9776.本文责任编委:李世华.国家自然科学基金项目(52170001)资助.Supported by the National Natural Science Foundation of China(52170001).1966控制理论与应用第40卷沉淀过程状态,据此调整加矾量,但是由于流动电流值和透光率是间接反应絮凝沉淀过程的相对值,而且对仪器的灵敏度和维护要求较高,影响了在实际应用中的效果.直接将沉淀池出水浊度作为被控变量来控制加矾量是目前加矾系统控制的主流.由于历史数据中包含了控制过程中的所有信息,数据驱动方法[4]可以通过对历史数据的训练获得控制器参数,数据驱动方法避免了传统控制方法对过程模型的依赖,但是历史数据信息的获取往往是不全面的,一定程度限制了数据驱动方法在实际应用中的推广.虽然絮凝沉淀过程难以精确建模,但仍然可以通过采用高级反馈控制和扰动估计等方法,对其过程模型不精确,以及水质、水量突变等因素作用下的扰动进行抑制.滑模控制(sliding mode control,SMC)是强非线性控制问题中的一种有效方法,具有抗扰动性强、动态响应快、控制实现简单等优势.目前,已有许多相关理论和应用研究[5–7].文献[5]针对一类非线性积分系统,利用有限时间控制技术,提出了一种输入饱和情况下的全局有限时间控制方案.文献[6]提出了一种新颖的二阶滑模(second-order sliding model,SOSM)控制方法来处理具有不匹配项的滑模动力学,从而减少控制通道中的项.文献[7]提出了一种带有有限时间扰动观测器(finite-time disturbance observer,FDOB)的连续动态滑模控制器.在实际应用中,SOSM控制使滑动变量的选择更加灵活,而且也更容易消除振颤问题. FDOB能够对扰动和过程不确定性进行估计,并通过前馈补偿设计减少对控制系统的不利影响.将扰动观测器与反馈控制相结合的复合控制方法是目前控制领域中抑制扰动和补偿模型不精确等问题的研究热点之一[8].本文提出了一种基于FDOB和SOSM的水厂加矾系统复合控制方法,针对实际絮凝沉淀过程受原水水质和水量突变的影响,以及强非线性、不确定性和参数时变等问题,采用前馈补偿和反馈控制相结合的设计方法.仿真结果证明,在与实际絮凝沉淀过程相符合的模型不匹配和扰动情况下,本文提出的控制方法更好地实现了出水浊度的稳定.2系统描述与控制器设计2.1问题描述自来水厂常规处理工艺流程如图1所示.其中,絮凝沉淀过程是在沉淀池入口处向原水中投加矾等絮凝剂,从而让各种杂质颗粒物等凝结成絮凝体,在重力作用下,絮凝体就能够沉淀在沉淀池底部,达到去浊澄清的目的.2.2控制器设计2.2.1SOSM已知系统状态等一些可测量的信息存在于˙s中,在˙s得到导数¨s的过程中可能会放大这些信息,所以需要更大的控制增益.由于振颤幅度与控制幅度之间存在正比关系,因此传统SOSM会导致振颤.图1自来水厂常规处理工艺流程Fig.1Conventional treatment process of waterworks针对上述问题,本文采用一种新的控制设计方法来处理具有不匹配不确定性的SOSM动力学.构造控制器包括3个步骤.首先,引入新的滑动变量,将传统SOSM动力学转变为具有不匹配不确定项的新型SOSM动力学.其次,通过定义失配不确定性的一些增长条件,以递归的方式构建一系列虚拟控制器来稳定新的滑动变量.最后,结合有限时间控制技术,设计一种带有非光滑项的SOSM控制器.本文通过将出水浊度与设定值的偏差作为输入,设计二阶滑模控制器.其中G(s)为G(s)=K(T1s+1)(T2s+1)=bs2+a1s+a0,(1)由上式可得¨x=−a1˙x−a0x+bu,(2)其中:x∈R n,代表出水浊度;u∈R,代表控制输入.现在将滑动变量s(即出水浊度误差)定义为s=x−x ref,其中x ref表示浊度设定值.二阶滑模动力学方程为{˙s1=s2,˙s2=a(t,x)+b(t,x)u+d(t),(3)其中:a(t,x)=−a1˙x−a0x,b(t,x)=b,d(t)=ξω1(t)+ω2(t),此处ω1(t)为模型不匹配不确定项,ω2(t)为模型匹配扰动项,ω1(t)与ω2(t)及其一阶导数是有界的,因此存在一个正常数D>0使得|d(t)| D.沿系统(3)对滑动变量进行二阶导数得到¨s= a(t,x)+ξω1(t)+ω2(t)+v,其中v=b(t,x)u.则系统(3)可以进一步表示为{˙s1=s2,˙s2=A(t,x)+U,(4)其中:U=v是一个虚拟控制器,A(t,x)=a(t,x)+ξω1(t)+ω2(t).在实际应用中,出水浊度x是有界的,这表示可以找到常数A0>0,使得|A(t,x)| A0.另外也存在正函数C(x)与正常数K m,使得|a(t,x)| C(x),b(t,x) K m.为简化表达式,定义⌊x⌋α=sgn x|x|α,∀x∈R,∀m>0.设计控制器[9]第11期王冬生等:基于有限时间扰动观测器的水厂加矾系统二阶滑模控制1967u=−C(x)K msgn(⌊s2⌋αr2+β1a r2⌊s1⌋αr1)−β2⌊⌊s2⌋αr2+β1a r2⌊s1⌋αr1⌋r3a.(5) 2.2.2FDOBFDOB是根据被控变量和控制变量对扰动进行估计的过程,将扰动估计作为前馈可以有效补偿扰动对被控过程的影响,从而达到抑制扰动的目的．给出的FDOB表示如下[10]:˙z0=v0,v0=−L1⌊z0−s1⌋23+z1,˙z1=v1+U,v1=−L2⌊z1−v0⌋12+z2,˙z2=−L3sgn(z2−v1),(6)其中:L1,L2和L3为正观测器增益,需要合理设计.然后,可以得到如下定理:定理1[11]如果FDOB构造为式(6),则不确定项A(t,x)可以在有限时间内通过Z2准确估计,即可以找到一个时刻T f>0使得z2≡A(t,x)对于∀t>T f.2.2.3FDOB-SOSM复合控制设计在FDOB和SOSM基础上设计复合控制方案,如图2所示.其中,FDOB采取主动抗扰动的策略对控制系统受到的外部扰动和模型不匹配进行估计进而抑制和消除.相较于只采用反馈控制,FDOB能更加有效地抑制干扰,极大地提高系统的鲁棒性.图2复合控制方案框图Fig.2Diagram of composite control scheme在滑模控制器设计中,令s1=s,s2=˙s,滑模动力学可以改写为˙s1=s2,˙s2=A(t,x)+U,此刻需要注意的是不确定项A(t,x)通常是不可测量的,在实际应用中关于A(t,x)的精确值是未知的,这表示控制器(5)不会直接运用于系统(4)中,为此,假设A(t,x)是可微,并且满足|˙A(t,x)|<L,其中L是一个Lipschitz 常数.构造一个FDOB来获取对不确定项A(t,x)的估计,并使用估计值ˆA(t,x)补偿不确定项A(t,x),估计值通常是有界的.3系统稳定性分析假设1存在一个正常数K m,一个正函数C(x),使得|a(t,x)| C(x),|b(t,x)|>K m和r1=2,r2= r1−τ,r3=r2−τ且τ∈(0,1].定理2在假设1下,有一个常数a r1和正函数β1(s1),β2(s1,s2),建立闭环系统(4)–(5)的有限时间稳定性.首先给出以下3个引理,然后给出定理2的证明.引理1[12]如果0<c 1,那么有以下不等式:∀x1∈R,x2∈R,|⌊x1⌋c−⌊x2⌋c| 21−c|⌊x1⌋−⌊x2⌋|c.引理2[13]如果a>0,b>0和实数c>0,那么有以下不等式:|x1|a|x2|b aa+bc|x1|a+b+ba+bc−a b|x2|a+b.引理3对于实数0<c 1,有以下不等式:对于x i∈R与i=1,···,n,(|x1|+···+|x n|)c |x1|c+···+|x n|c.证分步骤证明定理2.步骤1定义函数V1(s1)=r12ρ+τ|s1|2ρ+τr1,(7)且ρ a.然后鉴于假设1,V1(s1)沿SOSM动力学(4)的导数可以推导出为˙V1(s1)=⌊s1⌋2ρ−r2r1s2⌊s1⌋2ρ−r2r1(s2−s∗2)+⌊s1⌋2ρ−r2r1s∗2,(8)其中s∗2虚拟控制器,可以设计为s∗2=−β1(s1)⌊ξ1⌋r2a,(9)其中:ξ1=⌊s1⌋a r1,β1(s1) β0,β0>0.得出˙V1(s1) ⌊ξ1⌋2ρ−r2a(s2−s∗2)−β0⌊ξ1⌋2ρa.(10)步骤2定义函数V2(s1,s2)=V1(s1)+W2(s1,s2),(11)其中W2(s1,s2)可以设计为W2(s1,s2)=s2s∗2⌊⌊k⌋a r2−⌊s∗2⌋a r2⌋2ρ−r3a d k.(12) V2(s1,s2)沿系统(4)的导数由下式给出:˙V2(s1,s2)=˙V1(s1)+∂W2(s1,s2)∂s2˙s2+∂W2(s1,s2)∂s1˙s1,(13)可得出˙V2(s1,s2) −β0|ξ1|2ρa+⌊ξ1⌋2ρ−r2a(s2−s∗2)+∂W2(s1,s2)∂s2˙s2+∂W2(s1,s2)∂s1˙s1.(14)此时注意0<r ia1和|s2−s∗2|=|⌊s2⌋a r2·r2a−⌊s∗2⌋a r2·r2a|.(15)1968控制理论与应用第40卷使用引理2,可以从式(15)中计算出⌊ξ1⌋2ρ−r 2a (s 2−s ∗2) β023|ξ1|2ρa+c 2|ξ2|2ρa ,(16)其中c 2=r 22r 2a ρ(23−r 2a (2ρ−r 2)ρβ0)2ρ−r 2r2是一个正常数.同时,由引理1可以得知|∂W 2(s 1,s 2)∂s 1˙s 1|2ρ−r 3a |s 2−s ∗2||ξ2|2ρ−r 3a−1|∂⌊s ∗2⌋ar 2∂s 1˙s 1| γ1|ξ2|2ρ+τa −1|∂⌊s ∗2⌋a r 2∂s 1˙s 1|,(17)其中γ1=21−r 2a 2ρ−r 3a.通过式(9)可以得知⌊s ∗2⌋ar 2=⌊β1(s 1)⌋ar 2ξ1,(18)|∂⌊s ∗2⌋ar 2∂s 1| |∂βa r 21(s 1)∂s 1ξ1|+a r 1βa r 21(s 1)(|ξ1|+βa r 10(s 0)|ξ0|)1−r 1a ,(19)因为ξ0=0且使用引理3,还可以得出|s 2| |ξ2|r 2a+β1(s 1)|ξ1|r 2a.(20)通过将不等式(20)和系统(4)合并,使用引理2,可以得到两个正函数γ′1(s 1)和γ′2(s 1)使得|∂⌊s ∗2⌋ar 2∂s 1˙s 1| γ′1(s 1)|ξ1|1−τa +γ′2(s 1)|ξ2|1−τa .(21)将不等式(21)代入(17),并使用引理2,可以计算出正增益˜γ2(s 1)使得|∂W 2(s 1,s 2)∂s 1˙s 1| β022|ξ1|2ρa +β023|ξ1|2ρa +˜γ2(s 1)|ξ2|2ρa.(22)结合系统(4)得出∂W 2(s 1,s 2)∂s 2˙s 2=⌊⌊s 2⌋a r 2−⌊s ∗2⌋a r 2⌋2ρ−r 3a ˙s 2=⌊ξ2⌋2ρ−r 3a(A (t,x )+U ).(23)将不等式(16)(23)代入式(14)得到˙V 2(s 1,s 2) −β02|ξ1|2ρa +(c 2+˜γ2(s 1))|ξ2|2ρa +⌊ξ2⌋2ρ−r 3a(A (t,x )+U ).(24)根据不等式(24),可以设计U =−C (x )K mb (t,x )sgn ξ2−b (t,x )β2(s 1,s 2)⌊ξ2⌋r 3a,(25)且β2(s 1,s 2)c 2+˜γ2(s 1)+β02K m.此外s 2s ∗2⌊⌊k ⌋a r 2−⌊s ∗2⌋ar 2⌋2ρ−r 3ad k 21−r 2a|ξ2|2ρ+τa.(26)将控制器(25)代入式(24),结合V 1(s 1),可以验证出V 2(s 1,s 2) 2(|ξ1|2ρ+τa+|ξ2|2ρ+τa).通过使c =β02·22ρ2ρ+τ,可以证明˙V 2(s 1,s 2)+cV 2ρ2ρ+τ2(s 1,s 2) 0.(27)注意2ρ2ρ+τ∈(0,1).通过不等式(27),可以通过有限时间李雅普诺夫理论[14]得出闭环系统(4)(25)是全局有限时间稳定的.因此,闭环系统(4)–(5)实现了全局有限时间稳定性.证毕.然而,在实际应用中,无法使用FDOB 准确估计系统不确定项,始终存在观测误差|˜A(t,x )|=A (t,x )−z 2.因此,可以找到一个时刻T f 和一个正常数ε,使得|˜A(t,x )| ε,对于∀t T f .最后,结合SOSM 算法和FDOB 技术得到的最后一个结果由定理3给出.定理3在假设1下,有一个常数a r 1和正函数β1(s 1),β2(s 1,s 2)使得下面的SOSM 控制律成立:u =−C (x )K msgn(⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋a r 1)−β2(s 1,s 2)⌊⌊s 2⌋ar 2+βar 21(s 1)⌊s 1⌋ar 1⌋r 3a −z 2,(28)其中z 2是FDOB(6)给出的不确定项A (t,x )的估计,建立闭环系统(4)–(5)的有限时间稳定性.证根据U =v 和v =b (t,x )u 的定义,得到U =−C (x )K mb (t,x )sgn(⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1)−b (t,x )β2(s 1,s 2)×⌊⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1⌋r 3a −z 2,(29)将控制器(29)放入系统(4)中,可以得到{˙s 1=s 2,˙s 2=A (t,x )−z 2+U s ,(30)U s =−C (x )K mb (t,x )sgn(⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1)−b (t,x )β2(s 1,s 2)×⌊⌊s 2⌋a r 2+βa r 21(s 1)⌊s 1⌋ar 1⌋r 3a .(31)第11期王冬生等:基于有限时间扰动观测器的水厂加矾系统二阶滑模控制1969因为系统(30)与系统(4)结构相似,则系统(30)在控制器U s下将有限时间收敛到原点.由此进一步验证控制器不会在T f之前发散到无穷大.选择一个有限时间有限函数V(s1,s2)=12s21+12s22.(32)由于系统(30)中的不确定项A(t,x)总是有界的,因此可以很容易地得到˜A(t,x)=A(t,x)−z2也是有界的.因此,可以找到一个正常数Υ使得|˙s2| |˜A(t,x)|+|U s| Υ,(33)˙V(s1,s2)=s1s2+s2(A(t,x)−z2+U s)2V(s1,s2)+12Υ2,(34)之后可以得出结论V(s1,s2)=(V(s1(0),s2(0))+14Υ2)e2t−14γ2.(35)这意味着系统(30)的状态s1和s2在时间间隔(0, T f]内是有界的.此外,可以得出结论,系统(30)可以通过复合控制器(28)在有限域内稳定到原点.因此,滑动变量s可以在有限时间内稳定为零.证毕.4仿真验证加矾系统随着实际工况的变化而不同,本文采用在线辨识方法对加矾系统进行建模,即G(S)=1050s2+15s+1.(36)本文在MATLAB环境下进行控制仿真.模拟在0∼60min期间将出水浊度设定值保持在2NTU,在60∼120min期间将出水浊度设定值保持在1NTU.选择超调量、调节时间(∆=0.02min)和绝对误差积分(integral absolute error,IAE)作为量化指标来评估控制方案,即IAE(t)=1NN∑t=1|y r(t)−y(t)|,(37)其中:y r(t)是参考值,y(t)是实际过程输出.为了设计加矾系统的二阶滑模控制器,首先要选择一个滑动变量.将滑动变量s(即浊度误差)定义为s=y−y ref,(38)式中:y表示出水浊度,y ref表示出水浊度设定值,得到滑动变量s的动力学方程{˙s1=s2,˙s2=−0.3s2−0.02s1+0.2u.(39) 4.1模型不匹配情况在加矾系统中,由于天气恶劣或水源受到污染,原水水质有时会发生突变.这导致沉淀池的原水水质超出正常范围,并且建立的模型过程与实际过程不匹配.为了证实所提出的控制方案的鲁棒性,在模型不匹配的情况下,K和T提高20%,从而得到了传递函数G(S)=14.472s2+18s+1,(40)因此,滑动变量s的动力学方程{˙s1=s2,˙s2=−0.25s2−0.014s1+0.2u.(41)通过取−C(x)K m=−(1.5|s2|+0.1|s1|),α=2, r1=2,r2=1.6,r3=1.2,β1=0.4,β2=6控制器可以设计为u=−(1.5|s2|+0.1|s1|)sgn(⌊s2⌋21.6+0.421.6⌊s1⌋1)−6⌊⌊s2⌋21.6+0.421.6⌊s1⌋1⌋1.22.(42)为了更好展示FDOB-SOSM复合控制器的性能,仿真中将工业系统中广泛运用的比例–积分–微分(pr-oportional integral derivative,PID)控制器,SOSM控制器,FDOB-PID复合控制器加入对照实验中,仿真结果如图3和表1所示.由图3可以看出在0∼60min和60∼120min,本文提出的FDOB-SOSM复合控制,能够更好地跟踪出水浊度设定值(reference,REF)的变化;由表1可知FDOB-SOSM复合控制下的系统稳定时间最少,绝对误差积分最小,整体性能要优于其他控制器.3.02.52.01.51.00.50.0≤⍺/NTU020406080100120U / minFDOB + SOSMSOSMFDOB + PIDPIDREF图3模型不匹配情况仿真结果Fig.3Simulation results of model mismatch4.2受扰动情况在加矾系统中,由于原水水质和水量变化、以及传感器信号波动等原因会导致对加矾系统产生一定的扰动.因此考虑受扰动情况,由传递函数(36),得到滑动变量s的动力学方程{˙s1=s2,˙s2=−0.3s2−0.02s1+0.2u(t)+d(t),(43)1970控制理论与应用第40卷式中d (t )为扰动,仿真中取的是幅度0.05,频率为0.1的正弦信号.控制器采用式(42).表1模型不匹配情况控制性能指标Table 1Control performance index of model mismatch0∼60min60∼120min控制方案超调量/稳定绝对误超调量/稳定绝对误%时间/差积分/%时间/差积分/min NTU min NTUFDOB-SOSM 060.0994012.50.2045SOSM 114.50.11410160.2582FDOB-PID 6130.090336410.3036PID 27.535.50.18425049.50.4480仿真结果如图4和表2所示.由图4可以看出在0∼60min,只有FDOB-SOSM 控制方案很好的跟踪设定值.可以看出基于FDOB 的扰动估计补偿,使FDOB-SOSM 复合控制具有更好的抗扰动能力;同时,由表2可知FDOB-SOSM 复合控制下的系统稳定时间最少,绝对误差积分也最小.2.52.01.51.00.50.0≤⍺ / N T U020406080100120U / minFDOB + SOSM SOSMFDOB + PID PID REF图4受扰动情况仿真结果Fig.4Simulation results under disturbance表2受扰动情况控制性能指标Table 2Control performance index under disturbance0∼60min60∼120min控制方案超调量/稳定绝对误超调量/稳定绝对误%时间/差积分/%时间/差积分/min NTU min NTU FDOB-SOSM011.50.11020120.2144SOSM 8.5110.1153–>600.3158FDOB-PID –>600.1602–>600.3377PID–>600.1798–>600.41614.3模型不匹配受扰动情况为了进一步对比FDOB-SOSM 控制方案的性能,在模型不匹配且同时遭受扰动的情况下,由传递函数(40),得到滑动变量s 的动力学方程{˙s 1=s 2,˙s 2=−0.25s 2−0.014s 1+0.2u (t )+d (t ),(44)式中d (t )为扰动,仿真中取的是幅度0.05,频率为0.1的正弦信号,控制器采用式(42).仿真结果如图5和表3所示.由图5可以看出在0∼60min 和60∼120min,只有FDOB-SOSM 控制方案很好的跟踪设定值.可以看出模型不匹配和外部扰动时,基于FDOB 的扰动估计补偿,使FDOB-SOSM 复合控制具有更好的设定值跟踪和抗扰动能力.由表3可知,FDOB-SOSM 控制方案具有更好的鲁棒性、更快的响应和更小的超调.3.02.52.01.51.00.50.0≤⍺ / N T U020406080100120U / minFDOB + SOSM SOSMFDOB + PID PID REF图5模型不匹配受扰动情况仿真结果Fig.5Simulation results of model mismatch underdisturbance表3模型不匹配受扰动情况控制性能指标Table 3Control performance index of model mismatchunder disturbance0∼60min60∼120min控制方案超调量/稳定绝对误超调量/稳定绝对误%时间/差积分/%时间/差积分/min NTU min NTU FDOB-SOSM790.1018017.50.2332SOSM –>600.1623–>600.432FDOB-PID–>600.1498–>600.3833PID–>600.3109–>600.58185结论本文提出了一种水厂加矾系统的FDOB-SOSM 复合控制方案,采用了一种改进的带有非光滑项的SOSM 控制方法实现加矾反馈控制;FDOB 用于估计模型不匹配和扰动,并应用估计值作为前馈补偿削弱模型不匹配和扰动带来的不利影响.采用李亚普诺夫函数证明了系统的稳定性.在实际工程中存在的水第11期王冬生等:基于有限时间扰动观测器的水厂加矾系统二阶滑模控制1971质、水量突变等影响下造成的模型不匹配与扰动分别进行了仿真.仿真结果证明了控制方法的有效性.参考文献:[1]RATNAY AKA D D,BRANDT M J,JOHNSON M K.CHAPTER8-Water Filtration Granular Media Filtration.Oxford:Butterworth-Heinemann,2009.[2]CUI Fuyi,LI Guibai.Coagulation control technology offlowing cur-rent method.Water Supply and Drainage in China,1991,7(6):36–40.(崔福义,李圭白.流动电流法混凝控制技术.中国给水排水,1991, 7(6):36–40.)[3]LIU Qianjun,BAI Hua,LI Guibai.Intelligent control of light trans-mittance pulsatingflocculation dosing system.Water Supply and Drainage in China,2003,19(8):52–53.(刘前军,白桦,李圭白.透光率脉动絮凝投药系统的智能控制.中国给水排水,2003,19(8):52–53.)[4]AI Wei,ZHU Xuefeng.Data-driven direct control method for largelag process offlocculation and dosing in water plant.Control Theory &Applications,2011,28(3):335–342.(哀微,朱学峰.水厂絮凝投药大滞后过程的数据驱动直接控制方法.控制理论与应用,2011,28(3):335–342.)[5]DING Shihong,LI Shihua.Global Finite-time stabilization of nonlin-ear integral systems under input saturation.Acta Automatica Sinica, 2011,37(10):1222–1231.(丁世宏,李世华.输入饱和下的非线性积分系统的全局有限时间镇定.自动化学报,2011,37(10):1222–1231.)[6]DING S,LI S.Second-order sliding mode controller design subjectto mismatched term.Automatica,2017,77:388–392.[7]RAUF A,LI S,MADONSKI R,et al.Continuous dynamic slidingmode control of converter-fed DC motor system with high order mis-matched disturbance compensation.Transactions of the Institute of Measurement and Control,2020,42(14):2812–2821.[8]YANG Bo,SHU Hongchun,ZHU Dena,et al.Maximum powertracking sliding mode control of permanent magnet synchronous gen-erator based on disturbance observer.Control Theory&Applications,2019,36(2):207–219.(杨博,束洪春,朱德娜,等.基于扰动观测器的永磁同步发电机最大功率跟踪滑模控制.控制理论与应用,2019,36(2):207–219.)[9]LIU L,ZHENG W X,DING S.High-order sliding mode controllerdesign subject to lower-triangular nonlinearity and its application to robotic system.Journal of the Franklin Institute,2020,357(15): 10367–10386.[10]CHEN D,SD B,XW A,et posite SOSM controller for pathtracking control of agricultural tractors subject to wheel slip.ISA Transactions,2022,130:389–398.[11] A.LEV ANT.Higher-order sliding modes,differentiation and output-feedback control.International Journal of Control,2003:76(9/10): 924–941.[12]QIAN C,WEI L.A continuous feedback approach to global strongstabilization of nonlinear systems.IEEE Transactions on Automatic Control,2001,46(7):1061–1079.[13]BHAT S P,BERNSTEIN D S.Finite-time stability of continuousautonomous systems.SIAM Journal on Control and Optimization, 2000,38(3):751–766.[14]ZHU J,YU X,ZHANG T,et al.Sliding mode control of MIMOMarkovian jump systems.Automatica,2016,68:286–293.作者简介:王冬生博士,副教授,研究方向为人工智能、大数据处理及智能控制在水处理过程中的应用,E-mail:***********************.cn;张鹏硕士研究生,研究方向为智能控制在水处理过程中的应用,E-mail:*****************;孙锦昊本科生,研究方向为智能控制在水处理过程中的应用, E-mail:*****************;郭若寒本科生,研究方向为智能控制在水处理过程中的应用, E-mail:****************;蒋国平博士,教授,研究方向为复杂网络、复杂系统控制,E-mail: *****************.cn.。

ZHIYUN-v1.10 SMOOTH 4U s e r G u i d eContents■Product Introduction (1)■Get to Know Smooth 4 (2)■Battery and Charging Instructions (3)■Installation (4)■The Use of Stabilizer (6)■The Use of APP (13)■Calibration and Firmware Upgrade (15)■Basic Specifications (17)■Disclaimer and Warning (19)■Warranty Terms (24)■Contact Card (27)Product IntroductionThanks for using ZHIYUN products. ZHIYUN TM products bring you the fun of more professional shooting.As the latest flagship phone stabilizer born for filmmakers, Smooth 4 is equipped with diverse function keys. It can navigate camera parameter settings and realize zooming and focusing. Its four redesigned working and operating modes can capture more advanced footage. Smooth 4 will be your great and innovative camera assistant.Please read the user manual carefully before using the product.Product IntroductionSmooth 4 Stabilizer ×1Tripod ×1Packing ListThe product comes with the following items. In case that any item is found missing, please contact ZHIYUN or your local selling agent.Carrying Case × 1 USB Cable × 11. Roll Axis Thumb Screw2. Phone Clamp3. Mobile Charging Port4. Zoom/Focus Handwheel5. Control Panel (see more on Page 6)6. 1/4″ Threaded Hole7. Mobile Clamp Thumb Screw 8. Tilt Axis Motor 9. Roll Axis Motor 10. F ixing Buckle 11. P an Axis Motor 12. T ype-C USB Port 13. T rigger Button (see more on Page 6)Get to Know Smooth 4Get to Know Smooth 4Installation of PhonePush and hold open the camera clamp, slide the smartphone in as close as possible against the tilt axis motor, and clmap on. You can also loosen the Mobile Clamp Thumb Screw on the back and rotate the clamp to make the phone vertical.putting in the phone.Proper adjustment of gravity center ensures better power saving during operation. The stabilizer may still function well when the gravity center is not properly configured, however, it may lead to more power consumption on the motors, and can greatly affect the torque output.BalancingIf the smartphone fails to stay in level and keeps tilting when mounted, loosen the Roll (Y) Axis Thumb Screw to adjust the gravity center by sliding the horizontal arm (as marked red in the Figure) until the smartphone stays still and vertical on the tilt axis. Tighten the Thumb Screw after the smartphone is well balanced in level to ensure normal operations.The Introduction of Control Panelcamera focus and its zoom ratio.❸ Handwheel Zoom/Focus Switch Button [ ]●Press the button once to switch functions●When the button light is on, handwheel on the side controls the zoom of phone camera lens.●When the button light is off, handwheel on the side controls the focus of phone camera lens.❹ Parameter Display Button [ ]●Press the button once to display/close shooting parameters;●In album mode, press the button once to display photo parameters;●Long press the button to automatically return to “Full Auto Mode” on the APP.❺Resolution Ratio/Frame Rate Selection Button [ ]Press the button once to enter resolution ratio/frame rate adjustment menu.❻ Camera Switch Button [ ]Press the button once to switch between front and rear phone cameras.❼ Album Mode Button [ ]Press the button once to enter phone album and playback photos or videos.❽ Confirm/LED Light Button [ ]●The Button works as Confirm Button when pressed once;●Long Press the button to turn on/off Fill Light.❾compensation adjustment menu.❿ Thumb WheelRotate the Thumb Wheel to adjust current option parameters.⓫ Photo Button [ ]Single press to take pictures.⓬ Video Button [ ]Single press to start or stop recording.⓭ Mode Switch Button●Push slider up to enter PF(Pan Following) Mode;●Push slider down to enter L(Locking) Mode.⓮ Power Button [ ]Press the button for 2 seconds to turn on/off the stabilizer;⓯ Battery Level Indicator●The button shows how much power is left.●0~25%: one blue light;●25~50%: two blue lights;●50~75%: three blue lights;●75~100%: four blue lights.⓰ “PhoneGo” Mode Button - Full-speed Following Mode ButtonWhen pressing this button, the stabilizer enters Full-speed Following Mode - “PhoneGo”. Smooth 4 can then follow your move synchronously at full speed.⓱ Following Mode Button●When pressing the button, the stabilizer enters following mode;●When double pressing the button, the tilt axis motor and roll axis motor go back to initial state.1. I n standby mode, the stabilizer can still adjustphone camera parameters;2. P art of the function buttons mentionedabove can only be used when the stabilizer is connected to "ZY Play" App in the phone.follows handle to move in the horizontal direction.follow handle to move.Buttons onQuick Switch to Standby Mode While stabilizer is powered on, put the horizontal arm down by hand, the horizontal arm is automatically locked with Fixing Buckle and the stabilizer enters standby mode. Before turning on or waking up stabilizer, manually separates stabilizer from the Fixing Buckle.Moving horizontal arm to horizontal position to wake up stabilizer.Downloading APPDownload the APP from ZHIYUN'sofficial website www.zhiyun-tech.com, or by directly scanning the QRcode on the left (Android 5.0 above oriOS9.0 above required) to download.iOS or Android users can alsodownload the APP by searching "ZYPlay" in APP Store/Android Store.1. Y ou can make the most use of the functionsmatching the Smooth 4 tailor-made "ZY Play "APP.2. T he ZHIYUN APP is subject to regular update,try out now to discover more functions.How to connect1. Power on the stabilizer and open phonebluetooth.2. Open "ZY Play" APP and Tap "Connect YourDevice" to connect the stabilizerThe Introduction of Main FeaturesQuick Adjustment of Camera Parameter Control of Camera Video and Photo Control of Camera Focus and ZoomObject TrackingStabilizer ParameterSetting Time LapseStabilizer CalibrationPanoramic ShootingZY PlayWhen the stabilizer needs calibrationInitialization MethodAfter activating stabilizer and entering stand-by mode, put the stabilizer on the ground and wait for 30 seconds, then the initialization of stabilizer is finished.I f the angle deviation still persists after the initialization, please try again following the procedure.Six-side CalibrationSix-side calibration through APP:Please refer to “APP Operations” on Page 13 for more information; Connect the stabilizer to ZHIYUN’s APP - ZY Play - via Bluetooth, enter “calibration” and finish the six-side calibration following the APP instruction.S ix-side calibration is to make all six sides of the phone clamp vertical to the level surface.Firmware Upgrade Steptool and latest firmware corresponding to your stabilizer. (USB driver installation is not needed for Mac and WIN 10 system).2. C onnect the stabilizer to your computer via Micro USB cable, power on the stabilizer and finish the installation of the driver.3. P ower on the stabilizer and long press mode button to enter standby mode.4. E nter the downloaded “Zhiyun Gimbal Tools” and upgrade firmware following the instructions indicated in the Firmware Upgrade Tutorial.Motor Fine-tuningPlease refer to “APP Operations” on Page 13 for more information; Connect the stabilizer to ZHIYUN’s APP - ZY Play - via Bluetooth, enter “PTZ” and adjust the value of the tilt and roll axis angle to your actual needs.Min.Standard Max.Remarks Operation Vol.7.4VOperation Current120mA2500mA Charging Input Vol. 4.7V5V 5.5VCharging Input Current500mA-2000mAPower Output Vol.-5V-Power Output Current--1500mABuilt-in Battery Volume-2000mAh*2-Tilt Angle Range-240°-Roll Angle Range-240°-Pan Angle Range-300°-OperationTemperature-10℃25℃45℃Operation Time-12h-ExperimentalData1Charging Time- 3.5h-ExperimentalData2Charging EnvironmentTemperature-5℃-+60℃Payload75g-210g Horizontal ArmAdjustment Range-10mm-Clamp Range65mm-82mmApplication Sphere Any smart phone with width within therange of clampProduct model: SMA04"1" T his data is gathered when the temperature is 25℃ and the stabilizer is properly balanced. "2" S mooth 4 is protected and stops charging when the charging temperature is too high in order to protect device and batteries. When the temperature is 25℃, please use 5V/2A battery charger to proceed this test. The charging time varies according to different environment and actual results may be different.A ll the data in this guide is gathered from internal experiments of Zhiyun laboratories. Under different scenarios, data is different to some extent and please refer to actual use of Smooth 4.Thank you for using ZHIYUN Smooth 4. The information contains herein affects your safety and your legal rights and responsibilities. Read this entire document carefully to ensure proper configuration before use. Failure to read and follow the instructions and warnings herein may result in serious injury to you or bystanders, or damage to your device or property. ZHIYUN reserves the right of final explanation for this document and all relevant documents relating to Smooth 4, and the right to make changes at any time without notice. Please visit for the latest product information.By using this product, you hereby signify that you have read this document carefully and that you understand and agree to abide by the terms and conditions herein. You agree that you are solely responsible for your own conduct while using this product, and for any consequences thereof. You agree to use this product only for purposes that are proper and in accordance with all terms, precautions, practices, policies and guidelines ZHIYUN has made and may make available. ZHIYUN TM accepts no liability for damage, injury or any legal responsibility incurred directly or indirectly from the use of this product. Users shall observe safe and lawful practices including, but not limited to, those set forth herein.ZHIYUN TM is the trademark of Guilin Zhishen Information Technology Co., Ltd. (hereinafter referred to as "ZHIYUN" or "ZHIYUN TECH") and its affiliates. All product names or marks referred to hereunder are trademarks or registered trademarks of their respective holders.This guide is for reference only and does not constitute any kind of commitment. Product(s), including but not limited to its color, size, etc., are subject to the actual product.GlossaryThe following terms are used throughout the product literature to indicate various levels of potential harm when operating this product. WARNING: P rocedures, which if not properly followed, may incur property damage,grave accident, or serious injury. CAUTION: P rocedures, which if not properly followed, may incur property damageand serious injury.NOTICE: P rocedures, which if not properly followed, may incur property damage orminor injury.Reading TipSymbol Description: Operation and Use TipsImportant NotesData charges may occur when you scan QR code to read or download the electronic user manual online, so you’d better process under WIFI environment.WarningRead the ENTIRE User Guide to become familiar with the features of this product before operating. Failure to operate the product correctly can result in damage to the product or personal property and cause serious injury. This is a sophisticated product. It must be operated with caution and common sense and requires some basic mechanical ability. Failure to operate this product in a safe and responsible manner could result in injury or damage to the product or other property. This product is not intended for use by children without direct adult supervision. DO NOT use with incompatible components or in any way otherwise as mentioned or instructed in the product documents provided by ZHIYUN. The safety guidelines herein contain instructions for safety, operation and maintenance. It is essential to read and follow all of the instructions and warnings in the User Guide, prior to assembly, setup or use, in order to operate the product correctly and avoid damage or serious injury.Safe Operation Guidelines CAUTION1) S mooth 4 is a high-precision control device. Damage may be caused to Smooth 4 if it is dropped or subject to external force, and this may result in malfunction.2) M ake sure the rotation of the gimbal axes is not blocked by external force when Smooth 4 is turned on.3) S mooth 4 is not waterproof. Prevent contacts of any kind of liquid or cleaner with CRANE 2. It is recommended to use dry cloth for cleaning.4) P rotect Smooth 4 from dust and sand during use.WARNINGTo avoid fire, serious injury, and property damage, observe the following safety guidelines when using, charging, or storing your batteries.NOTICEBattery UseMake sure the batteries are fully charged before each time of use.Battery ChargingThe battery will stop charging automatically when it is full. Disconnect the batteries from the charger when fully charged.Battery Storage1. D ischarge the battery to 40%-65% if it will NOT be used for over 10 days. This can greatly extend the battery life.2. T he battery enters hibernation mode when voltage gets too low. Charge the battery will bring it out of hibernation.Please download the latest version from www. This document is subject to change without notice. ZHIYUN™ is a trademark of ZHISHEN. Copyright © 2019 ZHISHEN. All rights reserved.Warranty PeriodThis warranty does not apply to the followings1. C ustomers are entitled to replacement or free repair service in case of quality defect(s) found in the product within 15 days upon receipt of the product.2. C ustomers are entitled to free repair service from ZHIYUN for any product proven defective in material or workmanship that results in product failure during normal consumer usage and conditions within the valid warranty period, which is 12 months counting from the date of selling.3. S ome states or countries do not allow limitations on how long an implied warranty lasts, so the above warranty term may not apply to you.1. P roducts subjected to unauthorized repair, misuse, collision, neglect, mishandling, soaking, accident, and unauthorized alteration.2. P roducts subjected to improper use or whose labels or security tags have been torn off or altered.3. Products whose warranty has expired.4. P roducts damaged due to force majeure, such as fire, flood, lightening, etc.Warranty Claim Procedure1. I f failure or any problem occurs to your product after purchase, please contact a local agent for assistant, or you can always contact ZHIYUN’s customer service through email at service@ or website at www.zhiyun-tech. com.2. Y our local agent or ZHIYUN’s customer service will guide you through the whole service procedure regarding any product issue or problem you have encountered. ZHIYUN reserves the right to reexamine damaged or returned products.Customer informationCustomer Name :Contact No :Address :Sales InformationSales Date :Prod. Serial No :Dealer :Contact No :#1 Maintenance RecordService Date :Signature of Repairman :Cause of Problem :Service Result :□Solved□Unsolved□Refunded(Replaced)Contact CardWebsite Weibo Facebook YoutubeGoogle+Youku VimeoWechat Tel: +86 (0) 773-3561275USA Hotline: +1 808-319-6137,9:00-18:00 GMT-7,Mon-Fri Europe Hotline: 0031-297303057,10:00-17:00 GMT+1,Mon-Fri Web: E-mail:***********************Address: 6th floor Building No.13 Creative Industrial Park, GuiMo Road, Qixing District, Guilin。

毕业设计（论文）开题报告题目多旋翼无人机制导控制算法的设计专业名称电子信息科学与技术班级学号********学生姓名王华指导教师何矞填表日期2017年3月25日一、选题的依据及意义：在如今高科技的现代战场上，无人机已开始从传统地执行侦察、评估等战斗支援任务，向着具备杀伤能力的方向迅速发展，作为能够执行攻击和拦截任务的作战装备，成为影响作战进程的重要力量。

2001年10月份，美军在阿富汗战场上首次使用“捕食者”无人机对塔利班目标进行了实弹攻击，开创了无人机执行对地攻击任务的先例，在世界上造成了很大的轰动。

无人机的价值一般是有人驾驶战斗机的几分之一，相比有人驾驶飞机具有经济优势，而且由于无人机无需考虑人的生理因素，因而可以进行大过载机动，能有效地进行突防和进攻，所以无人机更有作战优势。

在最近几场局部战争中无人机被大量应用，在未来战场上的用途也将越来越多，成为世界各军事大国武器装备的发展重点。

就目前来说，无人机用于空战还有很长的一段路要走,而无人机用于执行对地攻击任务已经成为现实。

一般来说无人机执行对地攻击任务的主要过程为：在巡航过程中进行搜索探查，进而发现并识别目标，然后由火控系统发射武器对目标进行攻击。

在无人机对地攻击中，一般都使用精确制导武器，如制导炸弹或战术导弹。

制导武器经点火发射后，经过制导与控制，以一定的弹道接近目标，最终命中并毁伤目标。

综合火力/飞行控制[4]是在上世纪 70 年代中期由美国空军提出的一种航空技术概念。

它的基本思想是通过火力/飞行耦合器把火控系统与飞行控制系统结合起来，形成一个闭环的武器自动攻击、投放系统。

以自动实现瞄准攻击为目的，采用综合火力/飞行控制系统的作战飞机能够实现机载武器的自动化攻击，可缩短瞄准攻击时间，提高武器的命中率，同时也增加作战飞机的生存机会。

精确打击在当前和未来战场中具有举足轻重的地位，美军在其二十一世纪联合作战的纲领性文件中，明确提出了在信息时代，美军为掌握现代战争的主动权所必须具备的四种作战理念，其中的第二项便是精确打击。

3d MAX 菜单中英文对照表Absolute Mode Transform Type-in绝对坐标方式变换输入Absolute/Relative Snap Toggle Mode绝对/相对捕捉开关模式ACIS Options ACIS选项Activate活动;激活Activate All Maps激活所有贴图Activate Grid激活栅格；激活网格Activate Grid Object激活网格对象；激活网格物体Activate Home Grid激活主栅格;激活主网格ActiveShade实时渲染视图；着色；自动着色ActiveShade(Scanline）着色(扫描线）ActiveShade Floater自动着色面板;交互渲染浮动窗口ActiveShade Viewport自动着色视图Adaptive适配;自动适配；自适应Adaptive Cubic立方适配Adaptive Degradation自动降级Adaptive Degradation Toggle降级显示开关Adaptive Linear线性适配Adaptive Path自适应路径Adaptive Path Steps适配路径步幅；路径步幅自动适配Adaptive Perspective Grid Toggle适配透视网格开关Add as Proxy加为替身Add Cross Section增加交叉选择Adopt the File’s Unit Scale采用文件单位尺度Advanced Surface Approx高级表面近似；高级表面精度控制Advanced Surface Approximation高级表面近似；高级表面精度控制Adv。

Lighting高级照明Affect Diffuse Toggle影响漫反射开关Affect Neighbors影响相邻Affect Region影响区域Affect Region Modifier影响区域编辑器；影响区域修改器Affect Specular Toggle影响镜面反射开关AI Export输出Adobe Illustrator(＊.AI)文件AI Import输入Adobe Illustrator(＊.AI）文件Align对齐Align Camera对齐摄像机Align Grid to View对齐网格到视图Align Normals对齐法线Align Orientation对齐方向Align Position对齐位置(相对当前坐标系）Align Selection对齐选择Align to Cursor对齐到指针Allow Dual Plane Support允许双面支持All Class ID全部类别All Commands所有命令All Edge Midpoints全部边界中点;所有边界中心All Face Centers全部三角面中心;所有面中心All Faces所有面All Keys全部关键帧All Tangents全部切线All Transform Keys全部变换关键帧Along Edges沿边缘Along V ertex Normals沿顶点法线Along Visible Edges沿可见的边Alphabetical按字母顺序Always总是www_bitscn_com中国.网管联盟Ambient阴影色；环境反射光Ambient Only只是环境光；阴影区Ambient Only Toggle只是环境光标记American Elm美国榆树Amount数量Amplitude振幅;幅度Analyze World分析世界Anchor锚Angle角度;角度值Angle Snap Toggle角度捕捉开关Animate动画Animated动画Animated Camera/Light Settings摄像机/灯光动画设置Animated Mesh动画网格Animated Object动画物体Animated Objects运动物体;动画物体；动画对象Animated Tracks动画轨迹Animated Tracks Only仅动画轨迹Animation动画Animation Mode Toggle动画模式开关Animation Offset动画偏移Animation Offset Keying动画偏移关键帧Animation Tools动画工具Appearance Preferences外观选项Apply Atmospherics指定大气Apply—Ease Curve指定减缓曲线Apply Inverse Kinematics指定反向运动Apply Mapping指定贴图坐标Apply—Multiplier Curve指定增强曲线Apply To指定到；应用到Apply to All Duplicates指定到全部复本Arc弧；圆弧Arc Rotate弧形旋转;旋转视图；圆形旋转Arc Rotate Selected弧形旋转于所有物体;圆形旋转选择物；选择对象的中心旋转视图Arc Rotate SubObject弧形旋转于次物体；选择次对象的中心旋转视图Arc ShapeArc Subdivision弧细分；圆弧细分Archive文件归档Area区域Array阵列Array Dimensions阵列尺寸;阵列维数Array Transformation阵列变换ASCII Export输出ASCII文件Aspect Ratio纵横比Asset Browser资源浏览器Assign指定Assign Controller分配控制器Assign Float Controller分配浮动控制器Assign Position Controller赋予控制器Assign Random Colors随机指定颜色Assigned Controllers指定控制器At All Vertices在所有的顶点上At Distinct Points在特殊的点上At Face Centers 在面的中心At Point在点上Atmosphere氛围；大气层；大气，空气;环境Atmospheres氛围Attach连接;结合；附加Attach Modifier结合修改器Attach Multiple多项结合控制;多重连接Attach To连接到Attach To RigidBody Modifier连接到刚性体编辑器Attachment连接；附件Attachment Constraint连接约束Attenuation衰减AudioClip音频剪切板AudioFloat浮动音频Audio Position Controller音频位置控制器AudioPosition音频位置Audio Rotation Controller音频旋转控制器AudioRotation音频旋转Audio Scale Controller音频缩放控制器AudioScale音频缩放；声音缩放Auto自动Auto Align Curve Starts自动对齐曲线起始节点Auto Arrange自动排列Auto Arrange Graph Nodes自动排列节点Auto Expand自动扩展Auto Expand Base Objects自动扩展基本物体Auto Expand Children自动扩展子级Auto Expand Materials自动扩展材质Auto Expand Modifiers自动扩展修改器Auto Expand Selected Only自动扩展仅选择的Auto Expand Transforms自动扩展变换Auto Expand XYZ Components自动扩展坐标组成Auto Key自动关键帧Auto-Rename Merged Material自动重命名合并材质Auto Scroll自动滚屏Auto Select自动选择Auto Select Animated自动选择动画Auto Select Position自动选择位置bitsCN#com中国网管联盟Auto Select Rotation自动选择旋转Auto Select Scale自动选择缩放Auto Select XYZ Components自动选择坐标组成Auto—Smooth自动光滑AutoGrid自动网格；自动栅格AutoKey Mode Toggle自动关键帧模式开关Automatic自动Automatic Coarseness自动粗糙Automatic Intensity Calculation自动亮度计算Automatic Reinitialization自动重新载入Automatic Reparam.自动重新参数化Automatic Reparamerization自动重新参数化Automatic Update自动更新Axis轴;轴向;坐标轴Axis Constraints轴向约束Axis Scaling轴向比率Back后视图Back Length后面长度Back Segs后面片段数Back View背视图Back Width后面宽度Backface Cull背面忽略显示；背面除去;背景拣出Backface Cull Toggle背景拣出开关Background背景Background Display Toggle背景显示开关Background Image背景图像Background Lock Toggle背景锁定开关Background Texture Size背景纹理尺寸；背景纹理大小Backgrounds背景Backside ID内表面材质号Backup Time One Unit每单位备份时间Banking倾斜Banyan榕树Banyan tree榕树Base基本；基部；基点；基本色；基色Base/Apex基点/顶点Base Color基准颜色；基本颜色Base Colors基准颜色Base Curve基本曲线Base Elev基准海拔；基本海拔Base Objects导入基于对象的参数，例如半径、高度和线段的数目;基本物体Base Scale基本比率Base Surface基本表面;基础表面Base To Pivot中心点在底部Bevel Profile轮廓倒角Bevel Profile Modifier轮廓倒角编辑器;轮廓倒角修改器Bezier贝塞尔曲线Bezier Color贝塞尔颜色bbs。