二阶系统 滑模控制

二阶系统案例
二阶系统案例包括弹簧阻尼系统、一维物块的运动、二阶熵、二阶控制系统的性能等。

1. 弹簧阻尼系统是一个典型的二阶系统，其中k为弹簧系数，B为阻尼系数。

通过分析其方程，可以得到系统的动态响应性能，如响应的快速性和逼近预期响应的程度。

2. 一维物块的运动是另一个二阶系统的实例，其中物块的位置和速度作为状态变量。

通过设计滑模控制器，可以将物块控制到原点。

3. 二阶熵是一个用于描述系统混乱程度的概念，可以用来描述人工智能系统的混乱程度。

4. 二阶控制系统的性能方面包括单位脉冲函数的输入和阶跃响应等，这些性能可以通过计算相关参数如上升时间、峰值时间和超调量等来衡量。

综上所述，二阶系统在多个领域中都有广泛应用，可以通过分析其方程和性能参数来深入了解其动态行为和性能。

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.。

二阶切换线性系统的滑模控制Sliding Mode Control of Second-order Linear Switched SystemLIAO Yi-bo(China Nuclear Control System Engineering Co., Ltd, Beijing 100176, China): The problem about stability control of switched system as a difficulty in control field caused wide attention by the academia and industry. In this paper concerning about second-order switched linear system which the object characteristics are varied while switching happens. It could be considered as a special non-linear system, Basing on sliding mode control theory and related sliding mode controller could be designed. Finally, it shows that the effort and robustness of sliding mode controller is better than the controller designed using traditional PID controller by example's simulation validation and results comparing.1 概述切换动态系统是一类重要的混合动态系统，它是由几个连续时间子系统或离散时间子系统以及相应的切换规则构成。

二阶滑模控制（读书笔记）详细推导一、改进时间最优二阶滑模控制算法1、非线性系统()[100]x Ax B x u Df y x=++= 0()0()B x b x ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦010D ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦u 为系统的控制量输入电压，y 为台车输出转角，f 为转向负载和外界扰动之和，()b x 为系统的非线性控制增益。

2、选取滑模切换函数33222111()()()T d d d d s C x x x x c x x c x x =-=-+-+-采用极点配置或二次型最优法确定矢量C,保证系统进入滑动模态后具有满意的动态特性。

为构造s 的二阶趋近律,令12,y s y s ==，状态方程为122,y s y y s v ====当满足时间最优的目标时，可导出控制量v2222112211sgn ,022sgn(),02m m m m my y y y a y y a a v y y a y y a ⎧⎛⎫-++≠⎪ ⎪⎪⎝⎭=⎨⎪+=⎪⎩ 其轨迹由两段抛物线组成，v 的符号只切换一次，开关线为22102m y y y a +=，m a 为趋近滑模的最大加速度。

3、则 s 的一阶导数 ()[()]()T T d d T T T ds C x x C Ax B x u Df x C Ax b x u C Df C x =-=++-=++-其中12(,,1)T C c c =，0()0()B x b x ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦s 的二阶导数12()()[()]()()()()()(,,)()()(,,,)()T T dT T dT T T T dd d s C Ax b x u b x u C x C A Ax B x u Df b x u b x u C x C AAx C AB x u C ADf b x u b x u C x x x f x u b x u x x f u b x uψψψ=++-=++++-=++++-==++=+则控制量 11ˆˆˆˆˆˆ(())[(,,,)](())[(,,,)]d d u bx s x x f u b x v x x f u ψψ--=-=- 解得0()(0)()tu t u u d ττ=+⎰其中ˆψ是ψ相对应的标称值 把()u t 代入s 1ˆˆˆ(,,,)()(())[(,,,)]()()ˆˆ(,,,)(,,,)ˆˆ()()(,,,)()d d d d d s x x f u b x bx v x x f u b x b x x x f u x x f u v b x b x x x f u x vψψψψφξ-=+-=-+=∆+显然，式中(,,,)()d x x f u x φξ∆和是由外干扰和参数摄动引起的，理 想 情 况 下扰 动为零，可验证(,,,)=0d x x f u φ∆并且根据假设可以推出12(,,,)()d x x f u H r x r φξ∆≤≤≤,其中H 为正实数。

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 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