recently , there is much interest in the dynamics of traffic breakdowns behind bottlenecks . measurements of traffic breakdowns on various freeways in the usa , germany , , holland , and korea suggest that many dynamic aspects are universal and therefore accessible to a physical description .one common property is the capacity drop ( typically of the order of 20% ) associated with a breakdown , which leads to hysteresis effects and is the basis of applications like dynamic traffic control with the aim of avoiding the breakdown . in the majority of cases , traffic breaks down upstream of a bottleneck and the congestion has a stationary donstream front at the bottleneck .the type of bottleneck , e.g. , on - ramps , lane closings , or uphill gradients , seems not to be of importance .several types of congested traffic have been found , among them extended states with a relatively high traffic flow .these states , sometimes referred to as `` synchronized traffic '' , can be more or less homogeneously flowing , or show distinct oscillations in the time series of detector data . very often , the congested traffic flow is , apart from fluctuations , homogeneous near the bottleneck , but oscillations occur further upstream . in other cases ,one finds isolated stop - and - go waves that propagate in the upstream direction with a characteristic velocity of about 15 km / h . finally , there is also an observation of a traffic breakdown to a pinned localized cluster near an on - ramp .there are several possibilities to delineate traffic mathematically , among them macroscopic models describing the dynamics in terms of aggregate quantities like density or flow , and microsopic models describing the motion of individual vehicles .the latter include continuous - in - time models ( car - following models ) , and cellular automata .traffic breakdowns behind bottlenecks have been simulated with the non - local , gaskinetic - based traffic model ( gkt model ) , the khne - kerner - konhuser - lee model ( kkkl model ) , and with a new car - following model , which will be reported below .  for a direct comparison with empirical data, one would prefer car - following models . as the position and velocity of each car is known in such models, one can reconstruct the way how data are obtained by usual induction - loop detectors . to this end, one introduces `` virtual '' detectors recording passage times and velocities of crossing vehicles and compares this ouput with the empirical data .because traffic density is not a primary variable , this avoids the problems associated with determining the traffic density by temporal averages .the present study refers to publication , where , based on a gas - kinetic - based macroscopic simulation model , it has been concluded that there should be five different congested traffic states on freeways with inhomogeneities like on - ramps .the kind of congested state depends essentially on the inflow into the considered freeway section and on the `` bottleneck strength '' characterizing the inhomogeneity .this can be summarized by a phase diagram depicting the kind of traffic state as a function of these two parameters .a similar phase diagram has been obtained for the kkkl model .the question is , whether this finding is true for some macroscopic models only , or universal for a larger class of traffic models and confirmed by empirical data .  the relative positions of some of the traffic states in this phase space has been qualitatively confirmed for a korean freeway , but only one type of extended state has been measured there .furthermore , this state did not have the characteristic properties of extended congested traffic on most other freeways , cf .e.g. , .it is an open question to confirm the relative position of the other states .moreover , to our knowledge , there are no direct simulations of the different breakdowns using empirical data as boundary conditions , neither with microscopic nor with macroscopic models.  careful investigations with a new follow - the - leader model ( the idm model ) show that ( i ) the conclusions of ref are also valid for certain microscopic traffic models ( at least deterministic models with a metastable density range ) , ( ii ) the results can be systematically transferred to more general ( in particular flow - conserving ) kinds of bottlenecks , and a formula allowing to quantify the bottleneck strength is given [ see eq .( 15 ) ] , ( iii ) the existence of all predicted traffic states is empirically supported , and finally , ( iv ) all different kinds of breakdowns can be simulated with the idm model with _ empirically measured _ boundary conditions , varying only one parameter ( the average time headway t ) , which is used to specify the capacity of the stretch .the applied idm model belongs to the class of deterministic follow - the - leader models like the optimal velocity model by bando et al . , but it has the following advantages : ( i ) it behaves accident - free because of the dependence on the relative velocity , ( ii ) for similar reasons and because of metastability , it shows the self - organized characteristic traffic constants demanded by kerner et al . ( see fig .4 ) , hysteresis effects , and complex states , ( iii )all model parameters have a reasonable interpretation , are known to be relevant , are empirically measurable , and have the expected order of magnitude , ( iv ) the fundamental diagram and the stability properties of the model can be easily ( and separately ) calibrated to empirical data , ( v ) it allows for a fast numerical simulation , and ( vi ) an equivalent macroscopic version of the model is known , which is not the case for most other microscopic traffic models .these aspects are discussed in sec .ii , while sec .iii is not model - specific at all .section iii presents ways to specify and quantify bottlenecks , as well as the traffic states resulting for different traffic volumes and bottleneck strengths .the analytical expressions for the phase boundaries of the related phase diagram allow to conclude that similar results will be found for any other traffic model with a stable , metastable , and unstable density range .even such subtle features like tristability first found in macroscopic models are observed .it would be certainly interesting to investigate in the future , whether the same phenomena are also found for ca models or stochastic traffic models like the one by krauss .section iv discusses empirical data using representative examples out of a sample of about 100 investigated breakdowns .thanks to a new method for presenting the cross section data ( based on a smoothing and interpolation procedure ) , it is possible to present 3d plots of the empirical density or average velocity as a function of time _ and _ space .this allows a good imagination of the traffic patterns and a direct visual comparison with simulation results . in the idm microsimulations, we used a very restricted data set , namely only the measured flows and velocities at the upstream and downstream boundaries omitting the data of the up to eight detectors in between . although the simulated sections were up to 13 km long and the boundaries were typically outside of congestions , the simulations reproduced qualitatively the sometimes very complex observed collective dynamics .all in all the study supports the idea of the suggested phase diagram of congested traffic states quite well and suggests ways to simulate real traffic breakdowns at bottlenecks with empirical boundary conditions .for about fifty years now , researchers model freeway traffic by means of continuous - in - time microscopic models ( car - following models ) . since then , a multitude of car - following models have been proposed , both for single - lane and multi - lane traffic including lane changes .we will restrict , here , to phenomena for which lane changes are not important and only consider single - lane models . to motivate our traffic model we first give an overview of the dynamical properties of some popular microscopic models .continuous - time single - lane car - following models are defined essentially by their acceleration function . in many of the earlier models , the acceleration of vehicle , delayed by a reaction time , can be written as the deceleration is asumed to be proportional to the approaching rate of vehicle with respect to the leading vehicle .in addition , the acceleration may depend on the own velocity  and decrease with some power of the net ( bumper - to - bumper ) distance to the leading vehicle ( where is the vehicle length ) . since , according to eq .( [ gazis ] ) , the acceleration depends on a leading vehicle , these models are not applicable for very low traffic densities .if no leading vehicle is present ( corresponding to ) , the acceleration is either not determined or zero , regardless of the own velocity .however , one would expect in this case that drivers accelerate to an individual desired velocity .the car - following behavior in dense traffic is also somewhat unrealistic . in particular, the gap to the respective front vehicle does not necessarily relax to an equilibrium value .even small gaps will not induce braking reactions if the velocity difference is zero .these problems are solved by the car - following model of newell . in this model, the velocity at time depends adiabatically on the gap , i.e. , the vehicle adapts exactly to a distance - dependent function within the reaction time , the `` optimal velocity function '' \} ] on a free road and the tendency to brake with deceleration when vehicle comes too close to the vehicle in front .the deceleration term depends on the ratio between the `` desired minimum gap '' and the actual gap , where the desired gap is dynamically varying with the velocity and the approaching rate .  in the rest of this paper, we will study the case of identical vehicles whose model parameters , , , , , and are given in table [ tab : param ] . here , our emphasis is on basic investigations with models as simple as possible , and therefore we will set resulting in a model where all parameters have an intuitive meaning with plausible and often easily measurable values . while the empirical data presented in this paper can be nevertheless reproduced , a distinction of different driver - vehicle types and/or a nonzero  is necessary for a more quantitative agreement .a nonzero would also be necessary for features requiring an inflection point in the equilibrium flow - density relation , e.g. , for certain types of multi - scale expansions .special cases of the idm acceleration ( [ idmv ] ) with include the following driving modes :  [ [ equilibrium - traffic ] ] equilibrium traffic : + + + + + + + + + + + + + + + + + + + +  in equilibrium traffic of arbitrary density ( , ) , drivers tend to keep a velocity - dependent equilibrium gap to the front vehicle given by ^{-\frac{1}{2 } }        = ( s_0+v t )   \left [ 1 - \left(\frac{v}{v_0}\right)^{\delta}\right]^{-\frac{1}{2}}.\ ] ] in particular , the equilibrium gap of homogeneous _ congested _ traffic ( with ) is essentially equal to the desired gap , , i.e. , it is composed of a bumper - to - bumper space kept in standing traffic and an additional velocity - dependent contribution corresponding to a constant safe time headway .this high - density limit is of the same functional form as that of the newell model , eq .( [ newell ] ) . solving eq .( [ seq ] ) for leads to simple expressions only for , , or .in particular , the equilibium velocity for and is further interesting cases are and from a _ macroscopic _ point of view , equilibrium traffic consisting of identical vehicles can be characterized by the equilibrium traffic flow ( vehicles per hour and per lane ) as a function of the traffic density ( vehicles per km and per lane ) . for the idm model ,this `` fundamental diagram '' follows from one of the equilibrium relations ( [ seq ] ) to ( [ vedeltainfty ] ) , together with the micro - macro relation between gap and density : herein , the maximum density is related to the vehicle length by .figure [ fig : fund ] shows the fundamental diagram and its dependence on the parameters , , and . in particular , the fundamental diagram for and is identical to the equilibrium relation of the macroscopic gkt model , if the gkt parameter is set to zero ( cf .( 23 ) in ref . ) , which is a necessary condition for a micro - macro correspondence .  [ [ acceleration - to - the - desired - velocity ] ] acceleration to the desired velocity : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  if the traffic density is very low ( is large ) , the interaction term is negligible and the idm acceleration reduces to the free - road acceleration , which is a decreasing function of the velocity with a maximum value and . in fig . [fig : accbrake ] , this regime applies for times s. the acceleration exponent specifies how the acceleration decreases when approaching the desired velocity . the limiting case corresponds to approaching with a constant acceleration , while corresponds to an exponential relaxation to the desired velocity with the relaxation time . in the latter case ,the free - traffic acceleration is equivalent to that of the optimal - velocity model ( [ bando ] ) and also to acceleration functions of many macroscopic models like the kkkl model , or the gkt model .however , the most realistic behavior is expected in between the two limiting cases of exponential acceleration ( for ) and constant acceleration ( for ) , which is confirmed by our simulations with the idm . throughout this paperwe will use .[ [ braking - as - reaction - to - high - approaching - rates ] ] braking as reaction to high approaching rates : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  when approaching slower or standing vehicles with sufficiently high approaching rates , the equilibrium part of the dynamical desired distance , eq .( [ sstar ] ) , can be neglected with respect to the nonequilibrium part , which is proportional to .then , the interaction part of the acceleration equation ( [ idmv ] ) is given by this expression implements anticipative `` intelligent '' braking behavior , which we disuss now for the spacial case of approaching a standing obstacle ( ) .anticipating a constant deceleration during the whole approaching process , a minimum kinematic deceleration is necessary to avoid a collision .the situation is assumed to be `` under control '' , if is smaller than the `` comfortable '' deceleration given by the model parameter , i.e. , .in contrast , an emergency situation is characterized by . with these definitions ,( [ aapproach ] ) becomes while in safe situations the idm deceleration is less than the kinematic collision - free deceleration , drivers overreact in emergency situations to get the situation again under control .it is easy to show that in both cases the acceleration approaches under the deceleration law ( [ aaproach1 ] ) .notice that this stabilizing behavior is lost if one replaces in eq .( [ idmv ] ) the braking term by with corresponding to . the `` intelligent '' braking behavior of drivers in this regime makes the model collision - free .the right parts of the plots of fig .[ fig : accbrake ] ( s ) show the simulated approach of an idm vehicle to a standing obstacle .as expected , the maximum deceleration is of the order of . for low velocities ,however , the equilibrium term of can not be neglected as assumed when deriving eq .( [ aaproach1 ] ) .therefore , the maximum deceleration is somewhat lower than and the deceleration decreases immediately before the stop while , under the dynamics ( [ aaproach1 ] ) , one would have .similar braking rules have been implemented in the model of krauss , where the model is formulated in terms of a time - discretizised update scheme ( iterated map ) , where the velocity at timestep is limited to a `` safe velocity '' which is calculated on the basis of the kinematic braking distance at a given `` comfortable '' deceleration .[ [ braking - in - response - to - small - gaps ] ] braking in response to small gaps : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the forth driving mode is active when the gap is much smaller than but there are no large velocity differences .then , the equilibrium part of dominates over the dynamic contribution proportional to . neglecting the free - road acceleration , eq .( [ idmv ] ) reduces to , corresponding to a coulomb - like repulsion .such braking interactions are also implemented in other models , e.g. , in the model of edie , the gkt model , or in certain regimes of the wiedemann model .the dynamics in this driving regime is not qualitatively different , if one replaces by with .this is in contrast to the approaching regime , where collisions would be provoked for .figure [ fig : distance ] shows the car - following dynamics in this regime . for the standard parameters , one clearly sees an non - oscillatory relaxation to the equilibrium distance ( solid curve ) , while for very high values of ,the approach to the equilibrium distance would occur with damped oscillations ( dashed curve ) .notice that , for the latter parameter set , the _ collective _ traffic dynamics would already be extremely unstable .although we are interested in realistic _open _ traffic systems , it turned out that many features can be explained in terms of the stability behavior in a _ closed _ system .figure [ fig : stab](a ) shows the stability diagram of homogeneous traffic on a circular road .the control parameter is the homogeneous density .we applied both a very small and a large localized perturbation to check for linear and nonlinear stability , and plotted the resulting minimum ( ) and maximum ( ) densities after a stationary situation was reached .the resulting diagram is very similar to that of the macroscopic kkkl and gkt models . in particular, it displays the following realistic features : ( i ) traffic is stable for very low and high densities , but unstable for intermediate densities .( ii ) there is a density range of metastability , i.e. , only perturbations of sufficiently large amplitudes grow , while smaller perturbations disappear .note that , for most idm parameter sets , there is no second metastable range at higher densities , in contrast to the gkt and kkkl models .rather , traffic flow becomes stable again for densities exceeding the critical density , or congested flows below .( iii ) the density inside of traffic jams and the associated flow , cf .[ fig : stab](b ) , do not depend on .for the parameter set chosen here , we have vehicles / km , and , i.e. , there is no linearly stable congested traffic with a finite flow and velocity . for other parameters , especially for a nonzero idm parameter , both and can be nonzero and different from each other .  as further `` traffic constants '' , at least in the density range 20 veh./km 50 veh./km , we observe a constant outflow and propagation velocity km / h of traffic jams .figure [ fig : stab](b ) shows the stability diagram for the flows . in particular , we have , where , i.e. , the outflow from congested traffic is at the margin of linear stability , which is also the case in the gkt for most parameter sets . for other idm parameters ,the outflow  ] which can be used to determine and . only for nonzero ,one obtains an inflection point in the equilibrium flow - density relation .the acceleration coefficient influences the transition region between the free and congested regimes . for and ,the fundamental diagram ( equilibrium flow - density relation ) becomes triangular - shaped : /t) ] with km / h , , and the initial value . while this is considerably above minimum acceleration times ( 10 s - 20 s for average - powered cars ) , it should be characteristic for _ everyday _ accelerations .the comfortable deceleration m / s is also consistent with empirical investigations , and with parameters used in more complex models .  with efficient numerical integration schemes, we obtained a numerical performance of about vehicles in realtime on a usual workstation .road inhomogeneities can be classified into flow - conserving local defects like narrow road sections or gradients , and those that do not conserve the average flow per lane , like on - ramps , off - ramps , or lane closings .  _non - conserving _ inhomogeneities can be incorporated into macroscopic models in a natural way by adding a source term to the continuity equation for the vehicle density .an explicit microscopic modelling of on - ramps or lane closings , however , would require a multi - lane model with an explicit simulation of lane changes . another possibility opened by the recently formulated micro - macro link is to simulate the ramp section macroscopically with a source term in the continuity equation , and to simulate the remaining stretch microscopically .in contrast , _ flow conserving _ inhomogeneities can be implemented easily in both microscopic and macroscopic single - lane models by locally changing the values of one or more model parameters or by imposing external decelerations .suitable parameters for the idm and the gkt model are the desired velocity , or the safe time headway .regions with locally decreased desired velocity can be interpreted either as sections with speed limits , or as sections with uphill gradients ( limiting the maximum velocity of some vehicles ) .increased safe time headways can be attributed to more careful driving behavior along curves , on narrow , dangerous road sections , or a reduced range of visibility .local parameter variations act as a bottleneck , if the outflow from congested traffic ( dynamic capacity ) in the downstream section is reduced with respect to the outflow in the upstream section .this outflow can be determined from fully developed stop - and go waves in a _ closed _system , whose outflow is constant in a rather large range of average densities [ 20 veh./km , 60 veh./km ] , cf . fig .[ fig : stab](b ) .it will turn out that the outflow is the relevant capacity for understanding congested traffic , and not the maximum flow ( static capacity ) , which can be reached in ( spatially homogeneous ) equilibrium traffic only .the capacities are decreased , e.g. , for a reduced desired velocity or an increased safe time headway , or both .figure [ fig : qout ] shows and as a function of . for , traffic flow is always stable , and the outflow from jams is equal to the static capacity .for _ extended _ congested states , all types of flow - conserving bottlenecks result in a similar traffic dynamics , if is identical , where is the outflow for the changed model parameters , , etc .qualitatively the same dynamics is also observed in _ macroscopic _ models including on - ramps , if the ramp flow satisfies  .this suggests to introduce the following general definition of the `` bottleneck strength '' : in particular , we have for on - ramp bottlenecks , and for flow - conserving bottlenecks , but formula ( [ deltaq ] ) is also applicable for a combination of both .in contrast to _ closed _systems , in which the long - term behavior and stability is essentially determined by the average traffic density , the dynamics of _ open _ systems is controlled by the inflow to the main road ( i.e. , the flow at the upstream boundary ) .furthermore , traffic congestions depend on road inhomogeneities and , because of hysteresis effects , on the history of previous perturbations .  in this paper, we will implement flow - conserving inhomogeneities by a variable safe time headway .we chose as variable model parameter because it influences the flows more effectively than , which has been varied in ref .specifically , we increase the local safe time headway according to where the transition region of length is analogous to the ramp length for inhomogeneities that do not conserve the flow .the bottleneck strength  ] , one has a spatial _ coexistence _ hct / oct of states with hct near the bottleneck and oct further upstream .if , for yet higher flows , congested traffic is also convectively unstable , the resulting oscillations lead to tsg or oct . in summary ,the boundaries of the nonhysteretic transitions are given by congested traffic of the hct / oct type is frequently found in empirical data . in the idm , this frequent occurrence is reflected by the wide range of flows falling into this regime .for the idm parameters chosen here , we have , and vehicles / h , i.e. , _ all _ congested states are linearly unstable and oscillations will develop further upstream , while is nonzero for the parameters of ref . .free traffic is ( meta)stable in the overall system if it is ( meta)stable in the bottleneck region .this means , a breakdown necessarily takes place if the inflow exceeds the critical flow  ] , we set its velocity to that of the data , , if , and use the velocity and positional information of this vehicle to determine the acceleration of the vehicle behind .vehicle is taken out of the simulation as soon as vehicle has crossed the boundary .then , the velocity of vehicle is set to the actual boundary value , and so on .the downstream boundary conditions are only relevant for the time interval around h where traffic near this boundary is congested . for other time periods ,the simulation result is equivalent to using homogeneous von - neumann downstream boundary conditions .we modelled the stationary uphill bottleneck in the usual way by increasing the parameter to a constant value in the downstream region . the incident was already reflected by the downstream boundary conditions .figure [ flugzeug](c ) shows the simulation result in form of a spatio - temporal plot of the smoothed velocity .notice that , by using only the boundary conditions and a stationary bottleneck as specific information , we obtained a qualitative agreement of nearly all dynamical collective aspects of the whole complex scenario described above .in particular , for all times :50 h , there was free traffic at both upstream and downstream detector positions .therefore , the boundary conditions ( the detector data ) did not contain any explicit information about the breakdown to oct inside the road section , which nevertheless was reproduced correctly as an emergent phenomenon .finally , we consider a 30 km long section of the a5-north depicted in fig .[ plcmlcemp](a ) . on this section, we found one or more traffic breakdowns on six out of 21 days , all of them thursdays or fridays . on three out of 20 days , we observed one or more stop - and - go waves separated by free traffic .the stop - and go waves were triggered near an intersection and agreed qualitatively with the tsg state of the phase diagram .on one day , two isolated density clusters propagated through the considered region and did not trigger any secondary clusters , which is consistent with moving localized clusters ( mlc ) and will be discussed below . moving localized clusterswere observed quite frequently on this freeway section .again , they have a constant upstream propagation speed of about 15 km / h , and a characteristic outflow .in addition , we found four breakdowns to oct , and ten occurrences of pinned localized clusters ( plc ) .the plc states emerged either at the intersection i1 ( nordwestkreuz frankfurt ) , or 1.5 km downstream of intersection i2 ( bad homburg ) at cross section d13 .furthermore , the downstream fronts of all four oct states were fixed at the latter location .  on august 6 , 1998 , we found an interesting transition from an oct state whose downsteam front was at d13 , to a tsg state with a downstram front at intersection i2 ( d15 ) .consequently , we conclude that , around detector d13 , there is a stationary flow - conserving bottleneck with a stronger effect than the intersection itself .indeed , there is an uphill section and a relatively sharp curve at this location of the a5-north , which may be the reason for the bottleneck .the sudden change of the active bottleneck on august 6 can be explained by perturbations and the hysteresis associated with breakdowns .the different types of traffic breakdowns are consistent with the relative locations of the traffic states in the space of the phase diagram in fig .[ phasediag ] .three of the four occurrences of oct and two of the three tsg states were on fridays ( august 14 and august 21 , 1998 ) , on which traffic flows were about 5% _ higher _ than on our reference day ( friday , august 7 , 1998 ) , which will be discussed in detail below .apart from the coexistent plcs and mlcs observed on the reference day , all plc states occurred on thursdays , where average traffic flows were about 5% _ lower _ than on the reference day .no traffic breakdowns were observed on saturdays to wednesdays , where the traffic flows were at least 10% lower compared to the reference day . as will be shown below, for complex bottlenecks like intersections , the coexistence of mlcs and plcs is only possible for flows just above those triggering pure plcs , but below those triggering oct states .so , we have , with increasing flows , the sequence ft , plc , mlc - plc , and oct or tsg states , in agreement with the theory .now , we discuss the traffic breakdowns in august 7 , 1998 in detail .figure [ plcmlcemp](b ) shows the situation from 13:20 h until 17:00 h in form of a spatio - temporal plot of the smoothed density . during the whole time interval, there was a pinned localized cluster at cross section d13 . before :00 h , the plc state showed distinct oscillations ( oplc ) , while it was essentially stationary ( hplc ) afterwards .furthermore , two moving localized clusters ( mlc ) of unknown origin propagated through nearly the whole displayed section and also through a 10 km long downstream section ( not shown here ) giving a total of at least 30 km .remarkably , as they crossed the plc at d13 , neither of the congested states seemed to be affected .this complements the observations of ref . , desribing mlc states that propagated unaffected through intersections in the absence of plcs .as soon as the first mlc state reached the location of the on - ramp of intersection i1 ( km , 15:10 h ) , it triggered an additional pinned localized cluster , which dissolved at 16:00 h. the second mlc dissolved as soon as it reached the on - ramp of i1 at h.  figure [ plcmlcemp](c ) demonstrates that the mlc and plc states have characteristic signatures also in the empirical flow - density diagram .as is the case for hct and oct , the plc state is characterized by a two - dimensional flow - density regime ( grey squares ) .in contrast to the former states , however , there is no flow reduction ( capacity drop ) with respect to free traffic ( black bullets ) .as is the case for flow - density diagrams of hct compared to oct , it is expected that hplcs are characterized by an isolated region , while the points of oplc lie in a region which is connected to the region for free traffic . during the periodswere the mlcs crossed the plc at d13 , the high traffic flow of the plc state dropped drastically , and the traffic flow had essentially the property of the mlc , see also the velocity plot [ plcmlcemp](e ) .therefore , we omitted in the plc data the points corresponding to these intervals .  the black bullets for densities vehicles / km indicate the region of the mlc ( or tsg ) states .due to the aforementioned difficulties in determining the traffic density for very low velocities , the theoretical line given by ( see ref . and fig .[ plcmlcemp](b ) ) is hard to _ find empirically ._ in any case , the data suggest that the line would lie below the plc region .  to simulate this scenario it is important that the plc states occurred in or near the freeway intersections . because at both intersections , the off - ramp is upstream of the on - ramp [ fig . [plcmlcemp](a ) ] , the local flow at these locations is lower . in the following , we will investigate the region around i2 . during the considered time interval , the average traffic flow of both , the on - ramp and the off - ramp was about 300 vehicles per hour and lane . with the exception of the time intervals , during which the two mlcs pass by, we have about 1200 vehicles per hour and lane at i2 ( d15 ) , and 1500 vehicles per hour and lane upstram ( d16 ) and downstream ( d13 ) of i2 .this corresponds to an _ increase _ of the effective capacity by vehicles per hour and lane in the region between the off - ramp and the subsequent on - ramp .  in the simulation, we captured this qualitatively by _ decreasing _ the parameter in a section  ] , but as a capacity _ decrease _ for and .( for simplicity , we will not explicitely include the 400 m long transition regions of capacity increase at and decrease at in the following discussion . )then , the location can be considered as the beginning of a bottleneck , as in the system underlying the phase diagram .if the width of the region with locally increased capacity is _ larger than the width of plcs _ , such clusters are possible under the same conditions as in the standard phase diagram .in particular , traffic in the standard system is stable in regions upstream of a plc , which is the reason why any additional mlc , triggered somewhere in the downstream region and propagating upstream , will vanish as soon as it crosses the plc at .however , this disappearance is not instantaneous , but the mlc will continue to propagate upstream for an additional flow - dependent `` dissipation distance '' or `` penetration depths '' . if the width is _ smaller than the dissipation distance for mlcs _ , crossing mlcs will not fully disappear before they reach the upstream region .there , traffic is metastable again , so that the mlcs can persist . since , in the metastable regime , the outflow of mlcs is equal to their inflow ( in this regime , mlcs are equivalent to `` narrow '' clusters , cf . ) , the passage of the mlc does not change the traffic flow at the position of the plc , which can , therefore , persist as well .we performed several simulations varying the inflow within the range where plcs are possible . for smaller inflows ,the dissipation distance became smaller than , and the moving localized cluster was absorbed within the inhomogeneity .an example for this can be seen in fig .[ plcmlcemp](b ) at 16:40 h and 489 km .larger inflows lead to an extended oct state upstream of the capacity - increasing defect , which is also in accordance with the observations .in this paper , we investigated , to what extent the phase diagram fig , [ phasediag ] can serve as a general description of collective traffic dynamics in open , inhomogeneous systems .the original phase diagram was formulated for on - ramps and resulted from simulations with macroscopic models . by simulations with a new car - following model we showed that one can obtain the same phase diagram from microsimulations .this includes even such subtle details as the small region of tristability .the proposed intelligent - driver model ( idm ) is simple , has only a few intuitive parameters with realistic values , reproduces a realistic collective dynamics , and also leads to a plausible `` microscopic '' acceleration and deceleration behaviour of single drivers .an interesting open question is whether the phase diagram can be reproduced also with cellular automata .we generalized the phase diagram from on - ramps to other kinds of inhomogeneities .microsimulations of a flow - conserving bottleneck realized by a locally increased safe time headway suggest that , with respect to collective effects outside of the immediate neighbourhood of the inhomogeneity , all types of bottlenecks can be characterized by a single parameter , the bottleneck strength .this means , that the type of traffic breakdown depends essentially on the two control parameters of the phase diagram only , namely the traffic flow , and the bottleneck strength .however , in some multistable regions , the history ( i.e. , the previous traffic dynamics ) matters as well .we checked this also by macroscopic simulations with the same type of flow - conserving inhomogeneity and with microsimulations using a locally decreased desired velocity as bottleneck . in all cases, we obtained qualitatively the same phase diagram .what remains to be done is to confirm the phase diagram also for microsimulations of on - ramps .these can be implemented either by explicit multi - lane car - following models , or , in the framework of single - lane models , by placing additional vehicles in suitable gaps between vehicles in the `` ramp '' region .  by presenting empirical data of congested traffic, we showed that all congested states proposed by the phase diagram were observed in reality , among them localized and extended states which can be stationary as well as oscillatory , furthermore , moving or pinned localized clusters ( mlcs or plcs , respectively ) .the data suggest that the typical kind of traffic congestion depends on the specific freeway .this is in accordance with other observations , for example , moving localized clusters on the a5 north , or homogeneous congested traffic ( hct ) on the a5-south .in contrast to another empirical study , the frequent oscillating states ( oct ) in our empirical data did not show mergings of density clusters , although these can be reproduced with our model with other parameter values .the relative positions of the various observed congested states in the phase diagram were consistent with the theoretical predictions .in particular , when increasing the _ traffic flow _ on the freeway , the phase diagram predicts ( hysteretic ) transitions from free traffic to plcs , and then to extended congested states . by ordering the various forms of congestion on the a5-north with respect to the average traffic flow , the observations agree with these predictionsmoreover , given an extended congested state and increasing the _ bottleneck strength _, the phase diagram predicts ( non - hysteretic ) transitions from triggered stop - and - go waves ( tsg ) to oct , and then to hct . to show the qualitative agreement with the data, we had to estimate the bottleneck strength .this was done directly by identifying the bottleneck strength with ramp flows , e.g. , on the a5-north , or indirectly , by comparing the outflows from congested traffic with and without a bottleneck , e.g. , for the incident on the a5-south . with oct and tsg on the a5-north , but hct on the a5-south , where the bottleneck strength was much higher, we obtained again the right behavior .however , one needs a larger base of data to determine an empirical phase diagram , in particular with its boundaries between the different traffic states .such a phase diagram has been proposed for a japanese highway . besides plc states , many breakdowns on this freeway lead to extended congestions with fixed downstream _ and _ upstream fronts .we did not observe such states on german freeways and believe that the fixed upstream fronts were the result of a further inhomogeneity , but this remains to be investigated .our traffic data indicate that the majority of traffic breakdowns is triggered by some kind of _ stationary _ inhomogeneity , so that the phase diagram is applicable .such inhomogeneities can be of a very general nature .they include not only ramps , gradients , lane narrowings or -closings , but also incidents in the _ oppositely _ flowing traffic . in the latter case , the bottleneck is constituted by a temporary loss of concentration and by braking maneuvers of curious drivers _ at a fixed location_. from the more than 100 breakdowns on various german and dutch freeways investigated by us , there were only four cases ( among them the two moving localized clusters in fig .[ plcmlcemp ] ) , where we could not explain the breakdowns by some sort of stationary bottleneck within the road sections for which data were available to us .possible explanations for the breakdowns in the four remaining cases are not only spontaneous breakdowns , but also breakdowns triggered by a_ nonstationary _ perturbation , e.g. , moving `` phantom bottlenecks '' caused by two trucks overtaking each other , or inhomogeneities outside the considered sections .our simulations showed that stationary downstream fronts are a signature of non - moving bottlenecks .finally , we could qualitatively reproduce the collective dynamics of several rather complex traffic breakdowns by microsimulations with the idm , using empirical data for the boundary conditions .we varied only a _model parameter , the safe time headway , to adapt the model to the individual capacities of the different roads , _ and _ to implement the bottlenecks .we also performed separate macrosimulations with the gkt model and could reproduce the observations as well . because both models are effective - single lane models , this suggests that lane changes are not relevant to reproduce the collective dynamics causing the different types of congested traffic .furthermore , we assumed identical vehicles and therefore conclude that also the heterogeneity of real traffic is not necessary for the basic mechanism of traffic instability .we expect , however , that other yet unexplained aspects of congested traffic require a microscopic treatment of both , multi - lane traffic and heterogeneous traffic .these aspects include the wide scattering of flow - density data ( see fig .[ mixfund ] ) , the description of platoon formation , and the realistic simulation of speed limits , for which a multi - lane generalization of the idm seems to be promising . +* acknowledgments : * the authors want to thank for financial support by the bmbf ( research project sandy , grant no .13n7092 ) and by the dfg ( grant no .he 2789 ) .we are also grateful to the _ autobahndirektion sdbayern _ and the _ hessisches landesamt fr straen und verkehrswesen _ for providing the freeway data .m. cremer and f. meiner , _ traffic prediction and optimization using an efficient macroscopic simulation tool_. in : a. pave , ed ._ modelling and simulation 1993 _ , pp .513 - 519 .society for computer simulation international , ghent , belgium .speed limits lead to reduced desired velocities for many drivers as well , one might conclude that speed limits also cause undesirable local bottlenecks .the main effect of speed limits is , however , to reduce the velocity variance which leads to less braking maneuvers and _ de facto _ to a higher capacity .in contrast , gradients mainly influence slow vehicles ( trucks ) , leading to higher values for the velocity variance.      equilibrium flow - density relation of identical idm vehicles with ( a ) variable acceleration exponent , and ( b ) variable safe time headway and desired velocity .only one parameter is varied at a time ; the others correspond to the ones in table [ tab : param ] ., width=321 ]   temporal evolution of velocity and acceleration of a single driver - vehicle unit which accelerates on a 2.5 km long stretch of free road before it decelerates when approaching a standing obstacle at km .the dynamics for the idm parameters of table [ tab : param ] ( solid ) is compared with the result for an increased acceleration m / s ( dotted ) , or an increased braking deceleration m / s ( dashed ) . , title="fig:",width=321 ] +  temporal evolution of velocity and acceleration of a single driver - vehicle unit which accelerates on a 2.5 km long stretch of free road before it decelerates when approaching a standing obstacle at km .the dynamics for the idm parameters of table [ tab : param ] ( solid ) is compared with the result for an increased acceleration m / s ( dotted ) , or an increased braking deceleration m / s ( dashed ) . , title="fig:",width=321 ]   adaptation of a single vehicle to the equilibrium distance in the car - following regime .shown is ( a ) the net distance , and ( b ) the velocity of a vehicle following a queue of vehicles which all drive at km / h corresponding to an equilibrium distance m. the initial conditions are and m. the solid line is for the idm standard parameters , and the dashed line for the deceleration parameter increased from 1.67 m / s to 10 m / s . ,title="fig:",width=321 ] +  adaptation of a single vehicle to the equilibrium distance in the car - following regime .shown is ( a ) the net distance , and ( b ) the velocity of a vehicle following a queue of vehicles which all drive at km / h corresponding to an equilibrium distance m. the initial conditions are and m. the solid line is for the idm standard parameters , and the dashed line for the deceleration parameter increased from 1.67 m / s to 10 m / s . , title="fig:",width=321 ]   stability diagram of homogeneous traffic on a circular road as a function of the homogeneous density for small ( grey ) and large ( black ) initial perturbations of the density . in plot ( a ) , the upper two lines display the density inside of density clusters after a stationary state has been reached .the lower two lines represent the density between the clusters .plot ( b ) shows the corresponding flows and the equilibrium flow - density relation ( thin curve ) .the critical densities and flows are discussed in the main text . for vehicles/ km , the outflow and the corresponding density are constant . here, we have for the maximum equilibrium flow , but there are other parameter sets ( especially if ) where is clearly larger than  . ,title="fig:",width=321 ] +  stability diagram of homogeneous traffic on a circular road as a function of the homogeneous density for small ( grey ) and large ( black ) initial perturbations of the density . in plot ( a ), the upper two lines display the density inside of density clusters after a stationary state has been reached .the lower two lines represent the density between the clusters .plot ( b ) shows the corresponding flows and the equilibrium flow - density relation ( thin curve ) . the critical densities and flows are discussed in the main text . for vehicles/ km , the outflow and the corresponding density are constant . here, we have for the maximum equilibrium flow , but there are other parameter sets ( especially if ) where is clearly larger than  . , title="fig:",width=321 ]   outflow from congested traffic ( solid curve ) , minimum flow for nonlinear instabilities ( dashed ) , and maximum equilibrium flow ( dotted ) as a function of the safe time headway .an approximation for the bottleneck strength of the phase diagram is given by the difference between the value vehicles / h for s ( horizontal thin line ) , and .for decerasing values of traffic becomes more unstable which is indicated by increasing differences ( ) or ( ) . for this even leads to .furthermore , for s. , width=321 ]   phase diagram resulting from idm simulations of an open system with a flow - conserving bottleneck .the bottleneck is realized by an increased idm parameter in the downstream region , cf .( [ tx ] ) .the control parameters are the traffic flow and the bottleneck strength , see eq .( [ deltaq ] ) .the solid thick lines separate the congested traffic states tsg , oct , hct , plc , and mlc ( cf . the main text and fig . [ 3dphases ] ) and free traffic ( ft ) as they appear after adiabatically increasing the inflow to the value and applying a large perturbation afterwards ( history `` c '' in the main text ) . also shown is the critical downstream flow ( thin solid line ) , below which free traffic ( ft ) is ( meta-)stable , and the maximum downstream flow ( thin dotted ) below which ( possibly unstable ) equilibrium traffic exists .traffic is bistable for inflows above the lines ft - plc or ft - mlc ( whichever is lower ) , and below . in the smaller shaded region , traffic is tristable and the possible states ft , plc , or oct depend on the previous history ( see the main text ) . for history `` c ''we obtain oct in this region ., width=321 ]   spatio - temporal density plots of the traffic states appearing in the phase diagram of fig .[ phasediag ] .( a ) homogeneous congested traffic ( hct ) , ( b ) oscillating congested traffic ( oct ) , ( c ) triggered stop - and - go waves ( tsg ) , ( d ) ( stationary ) pinned localized cluster ( splc ) , and ( e ) oscillatory pinned localized cluster ( oplc ) .the latter two states are summarized as pinned localized clusters ( plc ) .after a stationary state of free traffic has developed , a density wave is introduced through the downstream boundary ( or initial conditions ) which eventually triggers the breakdown ( history `` c '' , cf . the main text ) ., title="fig:",width=321 ] +  spatio - temporal density plots of the traffic states appearing in the phase diagram of fig .[ phasediag ] .( a ) homogeneous congested traffic ( hct ) , ( b ) oscillating congested traffic ( oct ) , ( c ) triggered stop - and - go waves ( tsg ) , ( d ) ( stationary ) pinned localized cluster ( splc ) , and ( e ) oscillatory pinned localized cluster ( oplc ) .the latter two states are summarized as pinned localized clusters ( plc ) . after a stationary state of free traffichas developed , a density wave is introduced through the downstream boundary ( or initial conditions ) which eventually triggers the breakdown ( history `` c '' , cf . the main text ) ., title="fig:",width=321 ] +  spatio - temporal density plots of the traffic states appearing in the phase diagram of fig .[ phasediag ] .( a ) homogeneous congested traffic ( hct ) , ( b ) oscillating congested traffic ( oct ) , ( c ) triggered stop - and - go waves ( tsg ) , ( d ) ( stationary ) pinned localized cluster ( splc ) , and ( e ) oscillatory pinned localized cluster ( oplc ) .the latter two states are summarized as pinned localized clusters ( plc ) . after a stationary state of free traffichas developed , a density wave is introduced through the downstream boundary ( or initial conditions ) which eventually triggers the breakdown ( history `` c '' , cf .the main text ) ., title="fig:",width=321 ] +  spatio - temporal density plots of the traffic states appearing in the phase diagram of fig .[ phasediag ] .( a ) homogeneous congested traffic ( hct ) , ( b ) oscillating congested traffic ( oct ) , ( c ) triggered stop - and - go waves ( tsg ) , ( d ) ( stationary ) pinned localized cluster ( splc ) , and ( e ) oscillatory pinned localized cluster ( oplc ) .the latter two states are summarized as pinned localized clusters ( plc ) . after a stationary state of free traffichas developed , a density wave is introduced through the downstream boundary ( or initial conditions ) which eventually triggers the breakdown ( history `` c '' , cf . the main text ) ., title="fig:",width=321 ] +  spatio - temporal density plots of the traffic states appearing in the phase diagram of fig .[ phasediag ] .( a ) homogeneous congested traffic ( hct ) , ( b ) oscillating congested traffic ( oct ) , ( c ) triggered stop - and - go waves ( tsg ) , ( d ) ( stationary ) pinned localized cluster ( splc ) , and ( e ) oscillatory pinned localized cluster ( oplc ) .the latter two states are summarized as pinned localized clusters ( plc ) . after a stationary state of free traffichas developed , a density wave is introduced through the downstream boundary ( or initial conditions ) which eventually triggers the breakdown ( history `` c '' , cf . the main text ) ., title="fig:",width=321 ]   spatio - temporal density plots for the same phase point in the tristable traffic regime , but different histories .( a ) metastable plc and stable oct .the system is the same as in fig .[ 3dphases ] with vehicles/ h and s ( vehicles / h ) . the metastable plc is triggered by a triangular - shaped density peak in the initial conditions ( of total width 600 m , centered at ) , in which the density rises from 14 vehicles / km to 45 vehicles km .the oct is triggered by a density wave introduced by the downstream boundary conditions .( b ) same system as in ( a ) , but starting with metastable free traffic . here, the transition to oct is triggered by a density wave coming from the downstream boundary .( c ) similar behavior as in ( a ) , but for the gkt model ( with model parameters km / h , s , s , vehicles / km , , , and , cf . ref . ) . , title="fig:",width=321 ] +  spatio - temporal density plots for the same phase point in the tristable traffic regime , but different histories .( a ) metastable plc and stable oct .the system is the same as in fig .[ 3dphases ] with vehicles / h and s ( vehicles / h ) .the metastable plc is triggered by a triangular - shaped density peak in the initial conditions ( of total width 600 m , centered at ) , in which the density rises from 14 vehicles / km to 45 vehicles km .the oct is triggered by a density wave introduced by the downstream boundary conditions .( b ) same system as in ( a ) , but starting with metastable free traffic . here, the transition to oct is triggered by a density wave coming from the downstream boundary .( c ) similar behavior as in ( a ) , but for the gkt model ( with model parameters km / h , s , s , vehicles / km , , , and , cf . ref ., title="fig:",width=321 ] +  spatio - temporal density plots for the same phase point in the tristable traffic regime , but different histories .( a ) metastable plc and stable oct .the system is the same as in fig .[ 3dphases ] with vehicles/ h and s ( vehicles / h ) . the metastable plc is triggered by a triangular - shaped density peak in the initial conditions ( of total width 600 m , centered at ) , in which the density rises from 14 vehicles / km to 45 vehicles km .the oct is triggered by a density wave introduced by the downstream boundary conditions .( b ) same system as in ( a ) , but starting with metastable free traffic . here, the transition to oct is triggered by a density wave coming from the downstream boundary .( c ) similar behavior as in ( a ) , but for the gkt model ( with model parameters km / h , s , s , vehicles / km , , , and , cf ., title="fig:",width=321 ]   traffic breakdown to nearly homogeneous congested traffic on the freeway a5-south near frankfurt triggered by a temporary incident between 16:20 and 17:30 on aug . 6 , 1998 between the cross sections d11 and d12 .( a ) sketch of the freeway .( b ) flows at cross section d11 on the right lane ( solid black ) , middle lane ( grey ) , and left lane ( dotted ) ., title="fig:",width=321 ] +  traffic breakdown to nearly homogeneous congested traffic on the freeway a5-south near frankfurt triggered by a temporary incident between 16:20 and 17:30 on aug . 6 , 1998 between the cross sections d11 and d12 .( a ) sketch of the freeway .( b ) flows at cross section d11 on the right lane ( solid black ) , middle lane ( grey ) , and left lane ( dotted ) . , title="fig:",width=321 ]   * [ please display in two columns as indicated ] * details of the traffic breakdown depicted in fig .[ hctemp ] .( a ) flow - density diagram of the traffic breakdown on the a5 south , ( b)-(g ) temporal evolution of velocity and flow at three locations near the perturbation , and ( h ) , ( i ) inflow boundary conditions taken from cross section d6 . besides the data ( black lines ) ,the results of the microsimulation are shown ( grey ) .all empirical quantities are averages over one minute and over all three lanes ., width=604 ]   * [ please display in two columns as indicated ] * ( a ) , ( b ) smoothed spatio - temporal velocity and flow from the data of the traffic breakdown depicted in figs .[ hctemp ] and [ hctemptheo ] .( c ) , ( d ) corresponding idm microsimulation with the parameter set from table [ tab : param ] .the upstream boundary conditions for velocity and traffic flow were taken from cross section d6 . because of the fast relaxation of the velocity to the model s equilibrium value , the upstream boundary conditions for the empirical velocity plots ( a ) seem to be different from the simulation ( c ) ( see main text ) .homogeneous von neumann boundary conditions were assumed downstream .the temporary lane closing is modelled by locally increasing the idm parameter in a 1000 m long section centered at km during the time interval 16:20 h 17:30 h of the incident ., width=604 ]   * [ please display in two columns as indicated ] * traffic breakdown to oscillating congested traffic in the evening rush hour of october 29 , 1998 on the freeway a9-south near munich .( a ) sketch of the considered section with the cross sections d16 to d30 and their positions in kilometers .the small on- and off - ramps between i1 and i2 , which have been neglected in our simulation , are indicated by diagonal lines .( b ) spatio - temporal plot of the smoothed lane - averaged velocity .( c ) flow - density diagram obtained from two detectors in the congested region .( d)-(i ) lane - averaged 1-minute data of velocities and flows . , width=604 ]   smoothed spatio - temporal velocity plot from a qualitative idm microsimulation of the situation depicted in fig .[ octemp ] . as inflow boundary conditions, we used the traffic flow data of cross section d20 .homogeneous von neumann boundary conditions were assumed at the downstream boundary .the inhomogeneity was implemented by a local , but time - independent increase of in the region km from s to s. , title="fig:",width=359 ] +   oscillating congested traffic ( oct ) on an uphill section of the freeway a8 east ( near munich ) .( a ) sketch of the section with the cross sections d15 to d24 and their positions in kilometers .( b ) smoothed lane - averaged empirical velocity .an incident leading to a temporary lane closing between d23 and d24 ( near the downstream boundaries of the plots ) induces even denser congested traffic that propagates through the oct region .( c ) microsimulation using the data of cross sections d15 and d23 as upstream and downstream boundary conditions , respectively .the uphill section is modelled by linearly increasing the safe time headway from s ( for km ) to s ( for km ) . as in the previous microsimulations ,the _ velocity _ near the upstram boundary relaxes quickly and , therefore , seems to be inconsistent with the empirical values ( see main text ) ., title="fig:",width=321 ] +  oscillating congested traffic ( oct ) on an uphill section of the freeway a8 east ( near munich ) .( a ) sketch of the section with the cross sections d15 to d24 and their positions in kilometers .( b ) smoothed lane - averaged empirical velocity .an incident leading to a temporary lane closing between d23 and d24 ( near the downstream boundaries of the plots ) induces even denser congested traffic that propagates through the oct region .( c ) microsimulation using the data of cross sections d15 and d23 as upstream and downstream boundary conditions , respectively .the uphill section is modelled by linearly increasing the safe time headway from s ( for km ) to s ( for km ) . as in the previous microsimulations ,the _ velocity _ near the upstram boundary relaxes quickly and , therefore , seems to be inconsistent with the empirical values ( see main text ) ., title="fig:",width=340 ] +  oscillating congested traffic ( oct ) on an uphill section of the freeway a8 east ( near munich ) .( a ) sketch of the section with the cross sections d15 to d24 and their positions in kilometers .( b ) smoothed lane - averaged empirical velocity .an incident leading to a temporary lane closing between d23 and d24 ( near the downstream boundaries of the plots ) induces even denser congested traffic that propagates through the oct region .( c ) microsimulation using the data of cross sections d15 and d23 as upstream and downstream boundary conditions , respectively .the uphill section is modelled by linearly increasing the safe time headway from s ( for km ) to s ( for km ) . as in the previous microsimulations ,the _ velocity _ near the upstram boundary relaxes quickly and , therefore , seems to be inconsistent with the empirical values ( see main text ) ., title="fig:",width=340 ] +   data of two moving localized clusters ( mlcs ) , and pinned localized clusters ( plc ) on the freeway a5-north near frankfurt .( a ) sketch of the infrastructure with the positions of the cross sections d5 - d16 in kilometers .( b ) spatio - temporal plot of the density .( c ) flow - density diagram for detector d6 , where there are only stop - and - go waves ( ) , and for detector d13 ( ) , where we have omitted data points during the time intervals when the mlcs passed by .( d)-(f ) temporal evolution of the velocity ( one - minute data ) at the location of the plc at d13 , and upstream ( d18 ) and downstream ( d6 ) of it ., title="fig:",width=321 ] +  data of two moving localized clusters ( mlcs ) , and pinned localized clusters ( plc ) on the freeway a5-north near frankfurt .( a ) sketch of the infrastructure with the positions of the cross sections d5 - d16 in kilometers .( b ) spatio - temporal plot of the density .( c ) flow - density diagram for detector d6 , where there are only stop - and - go waves ( ) , and for detector d13 ( ) , where we have omitted data points during the time intervals when the mlcs passed by .( d)-(f ) temporal evolution of the velocity ( one - minute data ) at the location of the plc at d13 , and upstream ( d18 ) and downstream ( d6 ) of it ., title="fig:",width=359 ] +  data of two moving localized clusters ( mlcs ) , and pinned localized clusters ( plc ) on the freeway a5-north near frankfurt .( a ) sketch of the infrastructure with the positions of the cross sections d5 - d16 in kilometers .( b ) spatio - temporal plot of the density .( c ) flow - density diagram for detector d6 , where there are only stop - and - go waves ( ) , and for detector d13 ( ) , where we have omitted data points during the time intervals when the mlcs passed by .( d)-(f ) temporal evolution of the velocity ( one - minute data ) at the location of the plc at d13 , and upstream ( d18 ) and downstream ( d6 ) of it ., title="fig:",width=302 ] +  data of two moving localized clusters ( mlcs ) , and pinned localized clusters ( plc ) on the freeway a5-north near frankfurt .( a ) sketch of the infrastructure with the positions of the cross sections d5 - d16 in kilometers .( b ) spatio - temporal plot of the density .( c ) flow - density diagram for detector d6 , where there are only stop - and - go waves ( ) , and for detector d13 ( ) , where we have omitted data points during the time intervals when the mlcs passed by .( d)-(f ) temporal evolution of the velocity ( one - minute data ) at the location of the plc at d13 , and upstream ( d18 ) and downstream ( d6 ) of it ., title="fig:",width=302 ] +  data of two moving localized clusters ( mlcs ) , and pinned localized clusters ( plc ) on the freeway a5-north near frankfurt .( a ) sketch of the infrastructure with the positions of the cross sections d5 - d16 in kilometers .( b ) spatio - temporal plot of the density .( c ) flow - density diagram for detector d6 , where there are only stop - and - go waves ( ) , and for detector d13 ( ) , where we have omitted data points during the time intervals when the mlcs passed by .( d)-(f ) temporal evolution of the velocity ( one - minute data ) at the location of the plc at d13 , and upstream ( d18 ) and downstream ( d6 ) of it ., title="fig:",width=302 ] +  data of two moving localized clusters ( mlcs ) , and pinned localized clusters ( plc ) on the freeway a5-north near frankfurt .( a ) sketch of the infrastructure with the positions of the cross sections d5 - d16 in kilometers .( b ) spatio - temporal plot of the density .( c ) flow - density diagram for detector d6 , where there are only stop - and - go waves ( ) , and for detector d13 ( ) , where we have omitted data points during the time intervals when the mlcs passed by .( d)-(f ) temporal evolution of the velocity ( one - minute data ) at the location of the plc at d13 , and upstream ( d18 ) and downstream ( d6 ) of it . , title="fig:",width=302 ]   smoothed spatio - temporal plot of the traffic density showing the coexistence of a pinned localized cluster ( plc ) with moving localized clusters ( mlcs ) in an idm simulation .the plc is positioned at a road section with locally _ increased _ capacity corresponding to a bottleneck strength vehicles / h , which can be identified with the region between the off- and on - ramps of intersection i2 in fig . [ plcmlcemp](a ) .it was produced by locally _ decreasing _ the idm parameter from 1.9 s to s in a 400 m wide section centered at km , and increasing it again from to in a 400 m wide section centered at km .the initial conditions correspond to equilibrium traffic of flow vehicles / h , to which a triangular - shaped density peak ( with maximum density 60 vehicles / km at km and total width 1 km ) was superposed to initialize the plc .as upstram boundary conditions , we assumed free equilibrium traffic with a constant inflow of 1390 vehicles / h .as downstream boundary conditions for the velocity , we used the value for equilibrium free traffic most of the time .however , for two five - minute intervals at 14:20 h and 15:40 h , we reduced the velocity to km / h to initialize the mlcs .( b ) flow - density diagram of virtual detectors located at the position km ( plc ) , and 2.2 km downstream of it ( mlc ) .again , the time intervals , where density waves passed through the plc , were omitted in the ( `` virtual '' ) data points belonging to plc .the thin solid curve indicates the equilibrium flow density relation and the thick solid line characterizes the outflow from fully developed density clusters to free traffic , cf . , both for s. , title="fig:",width=396 ] +  smoothed spatio - temporal plot of the traffic density showing the coexistence of a pinned localized cluster ( plc ) with moving localized clusters ( mlcs ) in an idm simulation .the plc is positioned at a road section with locally _ increased _ capacity corresponding to a bottleneck strength vehicles / h , which can be identified with the region between the off- and on - ramps of intersection i2 in fig . [ plcmlcemp](a ) .it was produced by locally _ decreasing _ the idm parameter from 1.9 s to s in a 400 m wide section centered at km , and increasing it again from to in a 400 m wide section centered at km .the initial conditions correspond to equilibrium traffic of flow vehicles / h , to which a triangular - shaped density peak ( with maximum density 60 vehicles / km at km and total width 1 km ) was superposed to initialize the plc .as upstram boundary conditions , we assumed free equilibrium traffic with a constant inflow of 1390 vehicles / h . as downstream boundary conditions for the velocity, we used the value for equilibrium free traffic most of the time .however , for two five - minute intervals at 14:20 h and 15:40 h , we reduced the velocity to km / h to initialize the mlcs .( b ) flow - density diagram of virtual detectors located at the position km ( plc ) , and 2.2 km downstream of it ( mlc ) .again , the time intervals , where density waves passed through the plc , were omitted in the ( `` virtual '' ) data points belonging to plc .the thin solid curve indicates the equilibrium flow density relation and the thick solid line characterizes the outflow from fully developed density clusters to free traffic , cf . , both for s. , title="fig:",width=321 ]   flow - density diagram of a hct state of single - lane heterogeneous traffic consisting of 70% `` cars '' and 30% `` trucks '' .trucks are characterized by lower idm parameters and , and a larger compared to cars .the solid and dashed curves give the equilibrium flow - density relations for traffic consisting only of cars and trucks , respectively . for details , see ref ., width=283 ]

Abstract: we present data from several german freeways showing different kinds of congested traffic forming near road inhomogeneities , specifically lane closings , intersections , or uphill gradients . the states are localized or extended , homogeneous or oscillating . combined states are observed as well , like the coexistence of moving localized clusters and clusters pinned at road inhomogeneities , or regions of oscillating congested traffic upstream of nearly homogeneous congested traffic . the experimental findings are consistent with a recently proposed theoretical phase diagram for traffic near on - ramps [ d. helbing , a. hennecke , and m. treiber , phys . rev . lett . * 82 * , 4360 ( 1999 ) ] . we simulate these situations with a novel continuous microscopic single - lane model , the `` intelligent driver model '' ( idm ) , using the empirical boundary conditions . all observations , including the coexistence of states , are qualitatively reproduced by describing inhomogeneities with local variations of one model parameter . we show that the results of the microscopic model can be understood by formulating the theoretical phase diagram for bottlenecks in a more general way . in particular , a local drop of the road capacity induced by parameter variations has practically the same effect as an on - ramp .


model predictive control ( mpc ) for deterministic systems has received a considerable amount of attention over the last few decades , and significant advancements have been realized in terms of theoretical analysis as well as industrial applications .the motivation for such research thrust comes primarily from tractability of calculating optimal control laws for constrained systems .in contrast , the counterpart of this development for stochastic systems is still in its infancy .the deterministic setting is dominated by worst - case analysis relying on robust control methods .the central idea is to synthesize a controller based on the bounds of the noise such that a certain target set becomes invariant with respect to the closed - loop dynamics .however , such an approach usually leads to rather conservative controllers and to large infeasibility regions , and although disturbances are not likely to be unbounded in practice , assigning an a priori bound to them seems to demand considerable insight .a stochastic model of the disturbance is a natural alternative approach to this problem : the conservatism of the worst - case analysis may be circumvented , and one need not impose any a priori bounds on the maximum magnitude of the noise . however , since in practice control inputs are almost always bounded , it is of great importance to consider hard bounds on the control inputs as essential ingredients of the controller synthesis ; probabilistic constraints on the controllers naturally raise difficult questions on what actions to take when such constraints are violated ( see however  for one possible approach to answer these questions ) .in this paper we aim to provide answers to the following questions : given a linear system that is affected by ( possibly unbounded ) stochastic noise , to be controlled by applying predictive - type _ bounded _ control inputs , ( i ) is the associated optimization problem tractable ?( ii ) under what conditions is stability ( in a suitable stochastic sense ) of the closed - loop system guaranteed ?( iii ) is stability retained both in the case of mpc implementation and the case of rolling horizon control ( rhc ) implementation ?  in the deterministic setting , there exists a plethora of literature that settles tractability and stability of model - based predictive control , see , for example , and the references therein .however , there are fewer results in the stochastic case , some of which we outline next . in , the authors reformulate the stochastic programming problem as a deterministic one with bounded noise and solve a robust optimization problem over a finite horizon , followed by estimating the performance when the noise can take unbounded values , i.e. , when the noise is unbounded , but takes high values with low probability ( as in the gaussian case ) . in  a slightly different problem is addressed in which the noise enters in a multiplicative manner into the system , and hard constraints on the state and control input are relaxed to probabilistic ones .similar relaxations of hard constraints to soft probabilistic ones have also appeared in  for both multiplicative and additive noise inputs , as well as in  .there are also other approaches , for example those employing randomized algorithms as in  .finally , a related line of research can be found in , and a novel convex analysis dealing with chance and integrated chance constraints can be found in .  in this paperwe restrict attention to linear time - invariant controlled systems with affine stochastic disturbance inputs .our approach has three main features .firstly , for the finite - horizon optimal control subproblem we adopt a feedback control strategy that is affine in certain bounded nonlinear functions of the past noise inputs . secondly , instead of following the usual trend of adding element - wise constraints to the control input in the optimization , we propose a new approach that entails saturating the utilized noise measurements first and then optimizing over the feedback gains , ensuring that the hard constraints on the input will be satisfied by construction .this novel approach does not require artificially relaxing the hard constraints on the control input to soft probabilistic ones to ensure large feasible sets , and still provides a solution to the problem for a wide class of noise input distributions .in fact , we demonstrate that our strategy ( without state constraints ) leads to global feasibility .the effect of the noise appears in the finite - horizon optimal control problem as certain covariance matrices , and these matrices may be computed off - line and stored .thirdly , the measurement saturation functions are only required to be elementwise _ bounded _ in order to ensure tractability of the optimization problem while maintaining hard constraints on the control input ; therefore , these measurement saturation functions may be picked from among the wide class of saturation functions , the standard sigmoidal functions and their piecewise affine approximations , etc .once tractability of the finite - horizon underlying optimization problem is insured , it is possible to implement the resulting optimal solution using an mpc approach or an rhc approach . in the former case  ,the optimization problem is resolved at each step and only the first control input is implemented . in the latter case , the optimization problem is resolved every steps ( with being the horizon length ) and the entire sequence of input vectors is implemented . both of these approaches are shown to provide stability under the assumption that the zero - input and zero - noise system is asymptotically stable , which translates into the condition that the state matrix is schur stable . at a first glance, this assumption might seem restrictive .however , the problem of ensuring bounded variance of linear gaussian systems with bounded control inputs is , to our knowledge , still open , and here we are considering the problem of controlling a linear system with bounded control input and possibly unbounded noise .it is known that for discrete - time systems without any noise acting on the system it is possible to achieve global stability if and only if the matrix is neutrally stable  .this paper unfolds as follows . in [ s : ps ] we state the main problem to be tackled with the underlying assumptions . in [ s : if ] , we provide a tractable approach to the finite horizon optimization problem with hard constraints on the control input , as well as some examples in [ sec : examples ] . stability of the mpc and rhc implementations is shown in [ sec : stability ] , and hints onto the input - to - state stable properties of this result are provided in [ sec : iss ] . finally , we provide a numerical example in [sec : nexample ] and conclude in [ sec : conclusions ] .hereafter , is the set of natural numbers , , and is the set of nonnegative real numbers .we let denote the indicator function of a set , and and denote the -dimensional identity and zeros matrices , respectively .also , let  ] .  fix a horizon and set .the _ mpc _ procedure can be described as follows .* determine an admissible optimal feedback control policy , say , for an -stage cost function starting from time , given the ( measured ) initial condition ; * increase to , and go back to step ( a ) .  on the other hand , the _ rhc _ procedure simply replaces ( b ) above by  * apply the entire sequence of control inputs , update the state at the -th step , increase to and go back to step ( a ) .accordingly , the -th step of this procedure consists of minimizing the stopped -period cost function starting at time , namely , the objective is to find a feedback control policy that attains .\label{e : rhcf}\end{aligned}\ ] ] since both the system and cost are time - invariant , it is enough to consider the problem of minimizing the cost for , i.e. , the problem of minimizing over .+ in view of the above we consider the problem , \\ \textrm{s.t.}\quad&\mathrm{dynamics}\,(\ref{eq: system}),\ , \mathrm{and\,\ , constraints}\,(\ref{eq : bddu } ) \end{aligned}\ ] ] where and are some given symmetric matrices of appropriate dimension .if feasible with respect to , problem generates an optimal sequence of feedback control laws .the evolution of the system ( [ eq : system ] ) over a single optimization horizon can be described in compact form as follows : where } , \ ,\bar{w } : = \begin{bmatrix } w_0 \\ w_1 \\ \vdots \\ w_{n-1 } \end{bmatrix},\ ] ] }}\ ] ] where the input using the compact notation above , the optimization problem  can be rewritten as follows : , \\\textrm{s.t.}\quad & { \rm dynamics}\ , ( \ref{eq : compactdyn}),\ , \mathrm{and\,\ , constraints}\,(\ref{eq : bddu2 } ) , \\ \end{aligned}\ ] ] where },\ , \bar r={\left[\begin{matrix } r_0&\hdots & \mathbf 0_{m\times m } \\ \vdots & \ddots & \vdots\\  \mathbf 0_{m\times m } & \hdots & r_{n-1 }  \end{matrix}\right]}.\ ] ]  the solution to problem is difficult to obtain in general . in order to obtain an optimal solution to problem over the class of feedback policies , we need to solve the dynamic programming equationsthis generally requires using some gridding technique , making the problem extremely difficult to solve computationally .another approach is to restrict attention to a specific class of state feedback policies .this will result in a suboptimal solution to our problem , but may yield a tractable optimization problem .it is the track we pursue in the next section .by the hypothesis that the state is observed without error , one may reconstruct the noise sequence from the sequence of observed states and inputs by the formula in the light of this , and inspired by the works  , we shall consider feedback policies of the form : where the feedback gains and the affine terms must be chosen based on the control objective , while observing the constraints . with this definition ,the value of at time depends on the values of up to time . using ( [ eq : noiserec ] )we see that is a function of the observed states up to time .it was shown in  that there exists a one - to - one ( nonlinear ) mapping between control policies in the form ( [ eq : controlpolicy ] ) and the class of affine state feedback policies .that is , provided one is interested in affine state feedback policies , parametrization ( [ eq : noiserec ] ) constitutes no loss of generality .of course , this choice is generally suboptimal , but it will ensure the tractability of a large class of optimal control problems . in compact notation ,the control sequence up to time is given by where }{^\mathsf{t}} ] , is the -th row of the matrix , and is any function with . in other words ,we have chosen to saturate the measurements that we obtain from the noise input vector before inserting them into our control vector . this way we do not assume that the noise distribution is defined over a compact domain , which is an advantage over other approaches . moreover ,the choice of element - wise saturation functions is left open . as such , we can accommodate standard saturation , piecewise linear , and sigmoidal functions , to name a few .our choice of saturating the measurement from the noise vectors renders the optimization problem tractable as opposed to just calculating the whole input vector and then saturating it afterwards , which tends to an intractable optimization problem .note that the choices of control inputs in and are both _non markovian _ ; however , they differ in the fact that the former depends affinely on previous noise inputs , whereas the latter is a nonlinear feedback due to passing noise measurements through the function .[ p : main ] assume that =0 ] , we have that =\bar d ] , we have that = \ee_{\xz}\bigl[(\bar{g}\bar\varphi(\bar f\bar{w})+\bar{d}){^\mathsf{t}}m_1(\bar{g}\bar\varphi(\bar f\bar w)+\bar{d})\bigr ] { \nonumber}\\   & = \ee_{\xz}\bigl[\bar\varphi(\bar f\bar{w}){^\mathsf{t}}\bar g{^\mathsf{t}}m_1\bar g\bar\varphi(\bar f\bar{w})\bigr]+\bar d{^\mathsf{t}}m_1\bar d { \nonumber}\\   & = { \mathbf{tr}\!\left(\bar g{^\mathsf{t}}m_1\bar g \ee_{\xz}\bigl[\bar\varphi(\bar f\bar{w})\bar\varphi(\bar f\bar{w}){^\mathsf{t}}\bigr]\right)}+\bar d{^\mathsf{t}}m_1\bar d { \nonumber}\\   & = { \mathbf{tr}\!\left(\bar g{^\mathsf{t}}m_1\bar g\lambda_1\right)}+\bar d{^\mathsf{t}}m_1\bar d. \label{eq : arg2}\end{aligned}\ ] ] finally , combining and , we obtain the cost in problem [ eq : problem2 ] , which is convex .let us look at the constraints in problem [ eq : problem1 ] .the proposed control input satisfies the hard constraints as long as the following condition is satisfied : , such that .this is equivalent to the following conditions : , , such that .as these conditions should hold for any permissible value of the function , we can eliminate the dependence of the constraints on through the following optimization problems .it is straightforward now to show , using h " older s inequality , that , and the result follows .problem  is a quadratic program in the optimization parameters  , and can be solved efficiently by standard solvers such as ` cvx `  .an important step in the solvability of problem is being able to calculate the matrices and .in general , these matrices can be calculated off - line by numerical integration . however , in some instances these matrices can be given in terms of explicit formulas ; two of these instances are given in the following examples .recall the following standard special mathematical functions : the _ standard error function _ and the _ complementary error function _ defined by for , the _ incomplete gamma function _ defined by for , the _ confluent hypergeometric function _ defined by for and is the standard gamma function .we collect a few facts in the following  [ p : collect ] for we have  1 . ; 2 . + ; 3 . + ; 4 . ; 5 . .[ ex : sigmoids ] let us consider  with gaussian noise and sigmoidal bounds on the control input . more precisely ,suppose that the noise process is an independent and identically distributed ( i.i.d ) sequence of gaussian random vectors of mean and covariance .let the components of be mutually independent , which implies that is a diagonal matrix .suppose further that the matrix and that the function is a standard sigmoid , i.e. , .then from proposition [ p : collect ] we have for and ,        & = 2\cdot\frac{1}{\sqrt{2\pi}\sigma_i}\int_{0}^\infty \frac{t^2}{1+t^2 } { \mathrm e^{-\frac{t^2}{2\sigma_i^2}}}\\       & = \sqrt{2\pi}\sigma_i - \pi{\mathrm e^{-\frac{1}{2\sigma_i^2}}}\operatorname{erfc}\bigl(\frac{1}{\sqrt 2\sigma_i}\bigr ) .\end{aligned}\ ] ] this shows that the matrix in proposition [ p : main ] is equal to , where  similarly , since        & =  \frac{2}{\sqrt{2\pi}\sigma_i}\int_{-\infty}^\infty \frac{t^2}{\sqrt{1+t^2 } } { \mathrm e^{-\frac{t^2}{2\sigma_i } } } { \mathrm}dt\\       & = \frac{\sigma_i}{\sqrt 2}u\bigl(\frac{1}{2 } , 0 , \frac{1}{2\sigma_i^2}\bigr ) ,     \end{aligned}\ ] ] the matrix in proposition [ p : main ] is , where therefore , given the system , the control policy , and the description of the noise input as above , the matrices and derived above complete the set of hypotheses of proposition [ p : main ] .the problem  can now be solved as a quadratic program .note that we have chosen to use the standard sigmoidal functions in example [ ex : sigmoids ] .however , the result still holds for more general sigmoidal functions of the form , where is some given magnitude and is some given slope .this slight change is reflected in the entries of the matrices and , i.e. , for and , & = m\left(\sqrt{2\pi}\sigma_i\alpha - \pi{\mathrm e^{-\frac{1}{2\sigma_i^2\alpha^2}}}\operatorname{erfc}\left(\frac{1}{\sqrt 2\sigma_i\alpha}\right)\right ) ,     \end{aligned}\ ] ] and  = m\frac{\sigma_i \alpha}{\sqrt 2}u\bigl(\frac{1}{2 } , 0 , \frac{1}{2\sigma_i^2\alpha^2}\bigr) ] such that { \leqslant}\lambda v(x),\qquad \forall x\not\in k.\ ] ] then < \infty ] .we then have & = \ee_x\bigl[\ee\bigl[v(x_{t})\big|x_{t-1}\bigr]\bigr ] \\ & { \leqslant}\ee_x\bigl[\ee\bigl[\lambda v(x_{t-1 } ) + b{\mathbf 1_{k}}(x_{t-1})\bigr]\bigr]{\nonumber}\\       & { \leqslant}\lambda^t v(x ) + \sum_{i=0}^{t-1}\lambda^{t-1-i } b\;\ee_x\bigl[{\mathbf 1_{k}}(x_i)\bigr]{\nonumber}\\       & { \leqslant}\lambda^t v(x ) + \frac{b(1-\lambda^{t})}{1-\lambda } ,     \end{aligned}\ ] ] which shows that { \leqslant}v(x ) + b/(1-\lambda ) < \infty ] .  since by assumption the matrix is schur stable , there exists a positive definite and symmetric matrix with real entries , say , such that .using the system , at each time instant we have = \\ & \ee_{x_t}\bigl[(ax_t + b \bar d^*_{0|t } + fw_t+r){^\mathsf{t}}p(ax_t + b \bar d^*_{0|t } + fw_t+r)\bigr]\\         & = x_t{^\mathsf{t}}a{^\mathsf{t}}p a x_t + 2 x_t{^\mathsf{t}}a{^\mathsf{t}}p(b\bar d^*_{0|t}+f\mu_{w_t}+r ) \\ & \quad + \bar d^{*\mathsf t}_{0|t } b{^\mathsf{t}}pb\bar d^*_{0|t } + r{^\mathsf{t}}p r + 2(f\mu_{w_t}+r){^\mathsf{t}}pb \bar d^*_{0|t}\\ & \quad  + 2r{^\mathsf{t}}pf\mu_{w_t } + { \mathbf{tr}\!\left(f{^\mathsf{t}}pf\sigma_{w_t}\right)}.     \end{aligned}\ ] ] using the fact that ( from ) , we obtain the following bound & \leq  x_t{^\mathsf{t}}a{^\mathsf{t}}p a x_t + 2c_1{\left\lvert{x_t}\right\rvert}_\infty + c_2 ,     \end{aligned}\ ] ] where and . since , we have that \leq x_t{^\mathsf{t}}px_t - { \left\lvert{x_t}\right\rvert}^2 + 2c_1{\left\lvert{x_t}\right\rvert}_\infty + c_2 .     \end{aligned}\ ] ] for \max\{0,1-\lambda_{\max}(p)\ } , 1[ ] whence { \leqslant}\bigl(1- \frac{1-\theta}{{\lambda_{\text{max}}(p)}}\bigr)x_t{^\mathsf{t}}px_t,\,\quad\forall{\left\lvert{x_t}\right\rvert}_\infty > r.\ ] ] we see that the hypotheses of lemma [ l : fl ] are satisfied with , , and . since , it follows that { \leqslant}\frac{1}{\lambda_{\text{min}}(p)}\sup\limits_{t\in\nz}\ee_{\xz}\bigl[v(x_t)\bigr ] < \infty,\ ] ] which completes the proof .   in the rhc implementationis also iterative in nature , however instead of recalculating the gains at each time instant the optimization problem is solved every steps , where .the resulting optimal control policy ( applied over a horizon ) is given by , where again the control gains depend implicitly on the initial condition , i.e. , and .therefore , the optimal policy is given by .for , the resulting closed - loop system over horizon is given by where , and and are suitably defined matrices that are extracted from and , respectively .[ prop : rhmain ] assume that the matrix is schur stable and the assumptions of proposition [ p : main ] hold .then , under the control policy defined above , the closed loop system  satisfies < \infty ] , , we have that   =  x_{kn}{^\mathsf{t}}(a^\ell){^\mathsf{t}}p_\ell a^\ell x_{kn}\\ & \ , + 2x_{kn}{^\mathsf{t}}(a^\ell){^\mathsf{t}}p_\ell(\bar b_\ell \bar d^*_{kn}+\bar d_\ell\bar f\mu_{\bar w } + \bar d_\ell \bar r)+\bar r{^\mathsf{t}}\bar d_\ell{^\mathsf{t}}p_\ell \bar d_\ell \bar r\\&\ , + ( \bar d^*_{kn}){^\mathsf{t}}\bar b_\ell{^\mathsf{t}}p_\ell \bar b_\ell \bar d^*_{kn } + 2(\bar d^*_{kn}){^\mathsf{t}}\bar b_\ell{^\mathsf{t}}p_\ell\bar d_\ell(\bar f\mu_{\bar w}+\bar r)\\&\ , + 2\mu_{\bar w}{^\mathsf{t}}\bar f{^\mathsf{t}}\bar d_\ell{^\mathsf{t}}p_\ell\bar d_\ell\bar r  + { \mathbf{tr}\!\left((\bar g^*_{kn}){^\mathsf{t}}\bar b_\ell { ^\mathsf{t}}p_\ell \bar b_\ell \bar g_{kn}^*\lambda_1\right)}\\     & \ , + 2{\mathbf{tr}\!\left((\bar g^*_{kn}){^\mathsf{t}}\bar b_\ell{^\mathsf{t}}p_\ell\bar d_\ell\bar f\lambda_2\right ) } + { \mathbf{tr}\!\left(\bar f{^\mathsf{t}}d_\ell{^\mathsf{t}}p_\ell d_\ell \bar f \sigma_{\bar w}\right)}.{\nonumber}\end{aligned}\ ] ] using the fact that and ( from ) , we obtain the following bound \\ & \quad \leq  x_{kn}{^\mathsf{t}}(a^\ell){^\mathsf{t}}p_\ell a^\ell x_{kn } + 2 c_{1\ell } { \left\lvert{x_{kn}}\right\rvert}_\infty + c_{2\ell } ,   \end{aligned}\ ] ] where and  ] we know that where . from  it now follows that { \leqslant}x_{kn}{^\mathsf{t}}p_\ell x_{kn } - ( 1-\theta_\ell){\left\lvert{x_{kn}}\right\rvert}^2 , \forall { \left\lvert{x_{kn}}\right\rvert}_\infty > r_\ell ,    ] ,  ] for some than an estimate of  ] .the following proposition can be easily established with the aid of proposition [ prop : shmain ] and proposition [ prop : rhmain ] .the closed - loop systems  and  are input - to - state stable in .  the proof is omitted for space limitations .let us consider the system with some generic matrices } ] , , and .we simulate the system starting from different initial conditions , all of which are sampled according to a uniform distribution over ^ 3 $ ] .the noise inputs are independent and identically sampled according to a normal distribution , , the noise saturation function is chosen as in example [ ex : sigmoids ] with , and the input saturation bound .the optimization gain matrices are chosen to be and , , and the optimization horizon .the optimization matrices are given by , and .we used the ` cvx ` solver  to handle the optimization problem .the results for the mpc implementation are shown in figure [ fig : mpc ] , and those for the rhc implementation are shown in figure [ fig : rh ] , for the full state evolution over a horizon of 40 time steps .finally , it is interesting to note that the mpc and rhc _ average _ performance indices over the 50 different runs are given by 3985 and 4327 , respectively .in this paper , we provided a tractable optimization program that solves the stochastic model predictive control and rolling horizon control problems , while guaranteeing the satisfaction of hard bounds on the control input .we have showed that in both cases the resulting closed - loop process has bounded variance .we demonstrated that both implementations enjoy some qualitative notion of stochastic input - to - state stability .we provided several examples in which crucial matrices in our optimization program can be calculated off - line .future direction for this research is aimed at lifting the current feedback strategy onto general vector spaces .d. chatterjee , e. cinquemani , g. chaloulos , and j. lygeros , `` stochastic optimal control up to a hitting time : optimality and rolling - horizon implementation , '' 2008 .[ online ] .available : http://arxiv.org/abs/0806.3008          f. oldewurtel , c. jones , and m. morari , `` a tractable approximation of chance constrained stochastic mpc based on affine disturbance feedback , '' in _ conference on decision and control , cdc _ , cancun , mexico , dec .[ online ] .available : http://control.ee.ethz.ch / index.cgi?page = publications;action = details;id% = 3118[http://control.ee.ethz.ch / index.cgi?page = publications;action = details;id% = 3118 ]   m. maciejowski , a. lecchini , and j. lygeros , `` nmpc for complex stochastic systems using markov chain monte carlo , '' in _ international workshop on assessment and future directions of nonlinear model predictive control _ , ser .lecture notes in control and information sciences , vol .1em plus 0.5em minus 0.4emstuttgart , germany : springer , 2005 , pp .269281 .  a. a. stoorvogel , a. saberi , and s. weiland , `` on external semi - global stochastic stabilization of linear systems with input saturation , '' 2006 , submitted .[ online ] .available : http://homepage.mac.com/a.a.stoorvogel/subm03.pdf  m. agarwal , e. cinquemani , d. chatterjee , and j. lygeros , `` on convexity of stochastic optimization problems with constraints , '' in _european control conference _ , 2009 , submitted .[ online ] .available : http://control.ee.ethz.ch / index.cgi?page = publications;action = details;id% = 3271[http://control.ee.ethz.ch / index.cgi?page = publications;action = details;id% = 3271 ]         m. abramowitz and i. a. stegun , _ handbook of mathematical functions with formulas , graphs , and mathematical tables _ , ser. national bureau of standards applied mathematics series.1em plus 0.5em minus 0.4emfor sale by the superintendent of documents , u.s .government printing office , washington , d.c . , 1964 ,vol . 55 .v. s. borkar , `` uniform stability of controlled markov processes , '' in _ system theory : modeling , analysis and control ( cambridge , ma , 1999 ) _ , ser .kluwer international series in engineering computer science.1em plus 0.5em minus 0.4emboston , ma : kluwer academic publishers , 2000 , vol .107120 .j. spiliotis and j. tsinias , `` notions of exponential robust stochastic stability , iss and their lyapunov characterization , '' _ international journal of robust and nonlinear control _ , vol . 13 , no . 2 , pp .173187 , 2003 .r. m. dudley , _ real analysis and probability _ , ser .cambridge studies in advanced mathematics.1em plus 0.5em minus 0.4emcambridge : cambridge university press , 2002 , vol .74 , revised reprint of the 1989 original .

Abstract: this paper is concerned with the problem of model predictive control and rolling horizon control of discrete - time systems subject to possibly unbounded random noise inputs , while satisfying hard bounds on the control inputs . we use a nonlinear feedback policy with respect to noise measurements and show that the resulting mathematical program has a tractable convex solution in both cases . moreover , under the assumption that the zero - input and zero - noise system is asymptotically stable , we show that the variance of the state , under the resulting model predictive control and rolling horizon control policies , is bounded . finally , we provide some numerical examples on how certain matrices in the underlying mathematical program can be calculated off - line .