modeling and control design for a computer-controlled...

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 3, MAY 1997 279

Modeling and Control Design for aComputer-Controlled Brake System

Humair Raza, Zhigang Xu,Member, IEEE,Bingen Yang, and Petros A. Ioannou,Member, IEEE

Abstract—The brake subsystem is one of the most significantparts of a vehicle with respect to safety. A computer-controlledbrake system has the capability of acting faster than the humandriver during emergencies, and therefore has the potential ofimproving safety of the vehicles used in the future intelligenttransportation systems (ITS’s). In this paper we consider theproblem of modeling and control of a computer-controlled brakesystem. The brake model is developed using a series of exper-iments conducted on a test bench which contains the full scalebrake subsystem of a Lincoln town car and a computer-controlledactuator designed by Ford Motor Company. The developed modelhas the form of a first-order nonlinear system with the systemnonlinearities lumped in the model coefficients. The unknownmodel parameters are identified by applying curve fitting tech-niques to the experimental data. The major characteristics ofthe system such as static friction, dead zones, and hysteresishave been identified in terms of model parameters. The brakecontroller design makes use of a standard feedback linearizationtechnique along with intuitive modifications to meet the closed-loop performance specifications. The simulation results show thatthe proposed controller guarantees no overshoot and zero steady-state error for step inputs. Test of the same controller usingthe experimental bench setup demonstrates its effectiveness inmeeting the performance requirements.

Index Terms—Automated highway systems, automotive brakesystems, discrete-time control, modeling.

I. INTRODUCTION

W ITH an ever-increasing number of vehicles on thelimited highways, it has become urgent to develop

sophisticated technical solutions to today’s surface transporta-tion problems. Intelligent transportation systems (ITS’s) isan area whose purpose is to improve the efficiency of thecurrent transportation system through the use of advancedtechnologies. These technologies will be used to automate ve-hicles, infrastructure and improve the intelligence of decisionmaking. An important part of ITS is the advanced vehiclecontrol system (AVCS) whose purpose is to improve safetyand vehicular traffic flow rates by automating some or all ofthe basic functions of the vehicle, i.e., throttle, brake, andsteering control.

Manuscript received July 20, 1995; revised June 14, 1996. REcommendedby Associate Editor, H. Geering. This work was supported by the CaliforniaDepartment of Transportation through PATH of the University of Californiaand Ford Motor Company.

H. Raza and P. Ioannou are with the Department of Electrical EngineeringSystems, University of Southern California, Los Angeles, CA 90089-2563USA.

B. Yang is with the Department of Mechanical Engineering, University ofSouthern California, Los Angeles, CA 90089-1453 USA.

Z. Xu is with Northern Telecom, Richardson, TX 75235 USA.Publisher Item Identifier S 1063-6536(97)03272-7.

In this paper we consider the computer-controlled brake sub-system function. The objective is to understand the dynamicsof the braking function by modeling its behavior as a dynamicsystem and to design and test control algorithms for controllingit in order to meet the given performance requirements.

During the past few years, several attempts have beenmade by different research groups to develop models of thebrake subsystem for AVCS applications. One such importantcontribution is the work of Gerdeset al. [1]. A bond graphmethod for modeling the components of a manual brakesystem is considered in the paper by Khanet al. [2]. In thesestudies the emphasis was given on identifying the dynamicsassociated with each brake component. A comprehensivedynamic model of the brake subsystem for AVCS applications,which identifies the mapping from input to output, has notbeen addressed. The main purpose of this paper is to developa model and a controller for brake subsystem that can be usedin AVCS applications.

The brake model is developed using an experimental set upon a bench of the full scale Lincoln town car1 brake subsystem.The block diagram of the brake subsystem under study isshown in Fig. 1. The test bench has all the conventionalbrake components, and, in addition, it contains an auxiliaryhydraulic module (AHM) which consists of a hydraulic pump,control valves, and an actuator. It has been designed by Fordspecifically for automatic brake applications. The design usesbrake pedal actuation which is not as fast as some of the otheractuation techniques available, but has a nice feature of provid-ing driver override with minimal change in the conventionalbrake design. This actuator is currently being used for the im-plementation of intelligent cruise control (ICC) and is feasiblefor the initial design stages of ITS, where driver is still a part ofthe control loop. In these configurations the driver is ultimatelyresponsible for emergency braking and has the authority tooverride the automatic actuation at any time. The brake pedalactuation serves this purpose adequately and in our point ofview has a potential of being used in the initial design stages ofAVCS. The major drawback of large time delay can be reducedwith smart modifications in the current actuator design.

The dynamic system or model describing the input–outputbehavior of the brake subsystem is developed by using aseries of experiments and curve-fitting techniques to identifythe unknown parameters. The hysteresis has been modeled byidentifying two sets of parameters, and , for

1Lincoln town car is the trade mark of Ford Motor Company.

1063–6536/97$10.00 1997 IEEE

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280 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 3, MAY 1997

Fig. 1. Block diagram of the brake subsystem.

charging and discharging pressure modes declared as buildingand bleeding modes, respectively. The effect of dead zones andfriction on the system output has been quantitatively modeledas a percentage change in the system time constant. Theresulting model is a first-order nonlinear dynamic system thataccurately describes the dynamics of the brake subsystem.

The brake model is used to design a controller that canmeet the given performance and reliability requirements. Thecontroller employs feedback linearization to cancel the nonlin-earities and a modified proportional-integral (PI) compensatorto achieve the desired control action. The modeling and controltechniques used in this paper can be easily applied to othertypes of brake subsystems with minor modifications. Otherbrake control strategies which are used as part of vehiclelongitudinal controllers can be found in [3] and [4].

This paper is organized as follows. Section II describesbriefly the structure of the brake subsystem components. Fora more detailed descriptions the reader is referred to [1] and[5]. The brake subsystem model is developed in section III.In Section IV we consider the problem of identification of theunknown model parameters. This is followed by simulationresults and model validation. The control design with the mod-ification logic for the PI compensator is given in Section V.In Section VI the stability properties of the controller are dis-cussed briefly. The controller implementation and simulationresults are given in Section VII. This paper ends with the mainresults summarized in the conclusion section.

II. BRAKE SYSTEM COMPONENTS

The main components of the brake subsystem shown inFig. 1 are discussed below.

A. Auxiliary Hydraulic Module

The function of the AHM is to provide an input forceto the vacuum booster through an actuator and brake pedal.

The AHM takes the control input in the form of a pulsewidth modulated (PWM) signal and generates a pressure tobe applied to the brake pedal through an actuator. The PWMsignal is in the form of a square wave of fixed frequency butvarying duty cycle. The output pressure of the actuator andhence the brake line pressure can be controlled by changingthe duty cycle of the PWM signal.

As shown in Fig. 2, the AHM consists of a hydraulicpump, an arrangement of valves, and an actuator. When aconstant amount of fluid is pumped through the valves by thehydraulic pump, no pressure is developed inside the cylinderof the actuator if the valves are open, whereas a sudden riseof pressure is obtained if the valves are closed completely.Hence an average amount of pressure can be maintainedinside the cylinder of the actuator by switching the valves athigh frequency (typically 100 Hz) with changeable duty cycle(percentage of valve open time in one switching period). Thispressure pushes the piston of the actuator and applies force tothe brake pedal.

When the duty cycle is changed, the pressure inside thecylinder of the actuator changes too. Hence the force appliedto the brake pedal can be controlled by varying the duty cycleof the PWM signal. It should be noted that the maximum valueof duty cycle corresponds to valves being open for most of thetime and hence no force is applied to the brake pedal, whichresults in minimum brake line pressure at the output of themaster cylinder. From the overall system point of view anypermissible pressure value at the output of the master cylindercan be obtained by some particular value of duty cycle. Themodel developed in this paper identifies the mapping fromduty cycle to the line pressure.

B. Vacuum Booster

A simplified diagram of the vacuum booster is shownin Fig. 3. The force amplification is caused by a pressure


RAZA et al.: MODELING AND CONTROL DESIGN 281

Fig. 2. Block diagram of the auxiliary hydraulic module.

differential between the apply and vacuum chambers. Ideally,the amplification ratio between the input and output forcesshould be constant over the recommended range of operation.However, due to booster dynamics this ratio is not constant.According to the operation of the booster each brake appli-cation operation can be broken down into three basic stages:stage 1—apply; stage 2—hold or lap; and stage 3—release.These stages are shown in Fig. 4.

• In the apply stage control valve moves forward, theatmospheric valve is opened and the vacuum valve isclosed, hence a pressure differential is created, causingthe diaphragm to move forward.

• When the diaphragm travels further, the valve housingcatches up with the control valve. This movement alsocloses the atmospheric valve. The diaphragm and thevalve body are now in the hold stage.

• When the brake pedal is released, the control valve movesback due to the spring force, the apply and vacuumchambers are connected, and the pressure differential isreduced to zero.

Since the inertia of the push rod and diaphragm is quitesignificant, the associated dynamics can not be neglected.Furthermore, the changes of pressure in the apply and vacuumchambers also give rise to thermodynamics. For more detaileddiscussion of these effects, see [1].

C. Master Cylinder

The block diagram of a tandem master cylinder is shown inFig. 5. The input force, after being amplified by the vacuumbooster, is applied through a push rod to the primary piston.The secondary piston, however, is pushed by the hydraulicforce built up by the primary piston.

Each portion of the master cylinder has its own separatereservoir, compensating port, and outlet port. When an input

Fig. 3. Block diagram of the vacuum booster.

force is large enough to move the primary piston to close thecompensating port, pressure begins to build up between theprimary and secondary piston. When the secondary compen-sating port is closed, pressure buildup occurs in the secondaryportion too. At the same time hydraulic pressure developedduring this operation is transferred through the primary andsecondary brake lines to the brake pads.

As discussed in [1], since the masses of the pistons are neg-ligible, the dynamics associated with them can be neglected.

III. PROPOSEDBRAKE MODEL

A series of experiments are conducted on the test bench ofthe brake subsystem. Some of the results of the experimentswith step inputs of different magnitudes (corresponding toinputs of different duty cycle) are shown in Figs. 6 and7. These figures portray two basic modes of operation ofthe system: building and bleeding pressure modes. Anotherimportant system feature, which can be observed from these



Fig. 4. Vacuum booster operation.

figures, is the variable time delay associated with differentinputs and operating modes.

Since the time delay is an important factor in brakingoperations, a special attention was given to it in this study. Apure time delay (dead zone) of the order of 0.2 s is observedfor the relaxed system, i.e., when the line pressure is zero.This large time delay is reminiscent of the fact that in almostall of the hydraulic systems some control energy is requiredto overcome the static friction between the moving parts.Furthermore, the dead zones in vacuum booster and mastercylinder introduce an additional delay while the system isin relaxed state. However, once any measurable brake linepressure is observed, the subsequent inputs are transmittedthrough the system without any significant delay. Hence thetime delay becomes negligibly small 0.01 s) for any linepressure other than zero. This leads to the following relation:

ifelse

(1)

where and denote the delay time and brake line pressure,respectively. The system response in Fig. 8 shows the timedelays for the two cases discussed above.

The experimental results shown in Fig. 6 and 7 suggest thepresence of dominant first-order dynamics. These results to-gether with intuition motivate the following nonlinear dynamicmodel:

(2)

The variables in (2) are as follows:

system output (brake line pressure);system input (duty cycle of the PWM signal);unknown function to be identified.

is the time delay defined in(1) and is some small number (taken to be equal to thesampling period), hence denotes the previous input.As given in (1), the value of for which reflectsactuator dynamics, is quite small and can be neglected without



Fig. 5. Block diagram of master cylinder.

Fig. 6. Brake line pressure for building mode. Inputs range from duty cycle of 76–48%.

significant effect on modeling accuracy. Similarly, we willignore the dead zone identified in (1) and will account forits effect on the closed-loop stability by modification in thecontroller design. Hence can be safely assumed to be zeroin (2). Since in this study we use the input–output data whichis obtained at sampling instants only, instead of the continuousmodel in (2), we propose the discrete-time model

(3)

where denotes the number of the sample, i.e., the timeThe shape of the response for each fixed input, shown

in Figs. 6 and 7, can be approximated by a first-order systemgiven as

(4)

where The parameters and in (4) may beused to characterize the steady-state value and the speed of



Fig. 7. Brake line pressure for bleeding mode. The input is equal to 48% for0 � t< 30 s and is changed att = 30 s to different duty cyclevalues ranging from 50–90%.

(a)

(b)

Fig. 8. (a) For the relaxed system a large time delay, 0.2 s, is observed. (b) For any subsequent changes in the input, the time delay is very small (�0.01 s).

transient response, respectively, for a given input Theseparameters are nonlinear functions of the inputand state

of the system. The experiments show that the steady-state

value of the pressure can be modeled as

(5)



Fig. 9. Change inb for change in input from 56–50% (�u = 6) and from 68–64% (�u =4).

The speed of the transient response (time constant)variessignificantly for different initial conditions and is sensitive tothe previous history of the system, i.e., depends not only onthe current input but also on the previous inputThis phenomenon can be explained in terms of nonlinear fluiddynamics. The change in the pressure is significantly slow ifan input change occurs near a steady-state pressure condition.Hence the system time constant represented bydepends onthe current pressure relative to the steady-state pressure forthe previous input. This suggests a functional form for theparameter , which is given as

(6)

In (5) and (6), and are unknown functions to beidentified. We would like to comment here that the observednonlinearity in the brake system is a combination of friction,dead zones, and hysteresis. It is possible to identify thesenonlinearities independently, but in this paper we have used theapproach of modeling these phenomena through input–outputbehavior of the system. The parametersand in (4) areintroduced as the pseudo-steady state and a factor controllingthe dynamic behavior of the system, respectively. It is hardto identify the dependence of these parameters onto the actualnonlinearities of the system. These parameters, however, takeinto account the actual observed effects of these nonlinearities,such as the building and bleeding modes, time delay, anddependence of system dynamics on the initial condition of thesystem. The nonlinear dependence of these parameters on theinput and state of the system is shown through a stepwiseparameter identification process given below.

Fig. 10. Block diagram of the closed-loop system.

IV. PARAMETER IDENTIFICATION

The identification of the unknown parameters (functions)was done in two steps.

Step 1) Fixed step inputs were used to identify the steady-state value and time constant for each input.

Step 2) A series of staircase signals are used to modify theresults of Step 1.

The motivation of breaking down the identification processinto two steps follows from the fact that the system responseshown in Figs. 6 and 7, for a fixed input, can be approximatedby the response of a linear system to a step input. Hencestandard results from linear system identification can be usedto estimate the parameters. In the Step 2) the nonlinearbehavior of these parameters is explored by using staircasesignals. These signals cover the possible changes in the inputthat excite the building and bleeding modes of the system andenable us to study the switching process between these modes.The results from these experiments are used to modify theparameters and so that their values are valid for possibleinput variations applied to the system.



Fig. 11. Block diagram for implementation of the feedback linearized control system.

TABLE ISUMMARY OF CONTROL LAW

Finally, it is shown that with these parameters, identifiedby using step and staircase signals, a fairly accurate matchingbetween the model and the actual system can be achievedfor continuously varying inputs. This is due to the fact thatany continuous signal can be approximated by a staircasesignal, where the accuracy of the approximation depends onthe chosen step size. This approximation can be represented as

(7)

where is the sampling time and should be sufficiently smallcompared with the bandwidth of the input signal andsystem dynamics.

A. Step Inputs

The step response curves shown in Figs. 6 and 7 for eachfixed input can be approximated as a solution to the first-order linear differential equation, discretized at time stepi.e.,

(8)

where

steady-state value;time constant;initial condition.

This simplification helps us to identify the parametersandfor each input separately, by using standard-curve fitting

techniques. It should be noted that the approximation givenin (8) which uses a single exponential function does not

accurately describe the system response for inputs with dutycycle greater than 68% in the low-pressure region, whena significantly large time delay is observed as the energydeveloped by the actuator is barely sufficient to overcomethe friction and dead zones of the relaxed system. Hence thedeviation between the predicted system response, using themodel, and the actual system is large for small line pressures.However, line pressures below 60 psi, corresponding to theaforementioned inputs, have little or no significance in actualbraking. Hence this approximation has no effect on modelaccuracy within the range of line pressures of interest.

The steady-state value of the line pressurein the buildingmode is found to be relatively insensitive to the stateofthe system. Furthermore, the two modes of operation shownin Figs. 6 and 7 have different steady-state values and slopesfor the same inputs. The reason is the hysteresis produced dueto friction, preloaded spring inside the vacuum booster anddead zone associated with the master cylinder and booster.Hence separate mappings for the two modes are required. Theexperimental results in Fig. 6 suggest that for the buildingmode both and depend only on the current input, i.e.,

building (9)

On the other hand, for the bleeding modeis a function ofthe current input whereasdepends also on the current state

i.e.,

bleeding (10)

These mappings and are given in Tables III andIV, respectively.

B. Staircase Inputs

Another series of experiments was conducted with inputschanging from one value to a different value and these changeswere made to occur at different line pressures. Results showthat the values of calculated in these cases are consistentwith those given in Table III. The values of, however, varysignificantly and are found to be a function of the currentpressure at which the input was changed. Experiments showthat the change in the value of is noticeable if the inputis changed at a pressure which is more than 50% of thesteady-state value of the previous input. This change shows amonotonically decreasing behavior, with a maximum reductionof around 25% at a pressure approximately equal to thesteady-state value. It should be noted that these numbers were



Fig. 12. Block diagram of the modified integrator.

determined by an extensive series of tests on the system. It ishard to explain the physical significance of these numbers atthis time which were merely chosen to fit the observed data.However, the dependence of the system dynamicson theinitial state of the system can be thought of as the “springloading effect” in vacuum booster and master cylinder.

The change in the value ofas a function of the current linepressure for two different is shown in Fig. 9. Guided bythe experimental results, the following linear approximationwas introduced to model this change:

when

(11)

where

The experimental results also indicate that the change inoccurs only if the input changes, sincecorresponds to thedynamics associated with the system which do not significantlychange for a constant input. Whereas, the relation given in (11)updates the value of at each sampling instant, which can behandled by introducing as a state of the system as follows:

ifelse,

if

if

(12)

where and are design parametersto smooth out the effects of switching, which is justifiablesince the system dynamics do not show sudden changes. As

explained later, this filtering would also help in the controldesign.

From Table III, it is obvious that due to hysteresis,. From the experiments it was found that

if a change in the input causes the system state to be switchedfrom building to bleeding mode withthen , and the line pressure would maintain itsprevious value. Hence for the bleeding modein (10) canbe rewritten as

(13)

The condition for determining the current mode of operationis

pressure is building (14)

This means that if the current pressure is strictly less than thesteady-state value for the current input, then the system is inthe building mode. On the other hand if the current pressure isgreater than or equal to the steady-state pressure for the currentinput, then the system is in the bleeding mode. Hence by usingthe condition (14) the results given in (9), (10), (12), and (13)can be combined to give the final form of the model as

ifif

ifelse

if

if

if

(15)



Fig. 13. Block diagram of the feedback linearized control system.

(a)

(b)

Fig. 14. (a) Comparison of the actual and model output for a step input of 52%. (b) Input changes from 48–90% att =30 s.

The model described by (15) was simulated for different inputsand its response compared with that of the actual system. Theresults are shown in Figs. 14 and 15. From Fig. 15, we seethat the actual system and model output differ for low pressurevalues 60 psi). However, as discussed before, due to the lesssignificant effects of these pressure values on actual braking,this error is not severe.

It can also be seen from Fig. 15 that for pressure valuesgreater than 60 psi the error is within8 psi. These errorvalues are not large considering the fact that the pressuresensor used in the actual system has resolution of 4 psi.Furthermore, from Fig. 15 it is obvious that the model outputfor staircase approximation of continuous signals is the sameas that of the original signal. Hence, the estimates of the



(a) (b)

(c) (d)

Fig. 15. (a) The input signal being applied to actual system and brake model for comparison. (b) The system and model outputs for given input signal.(c) A staircase approximation of the same input signal. (d) The comparison of outputs.

TABLE IICOMPARISON OF PERFORMANCE

parameters and which were calculated as though the inputsignal is made up of finite steps hold even when the stepwidth is reduced to zero.

The limitations of the model (15) are the following.

1) The model accuracy depends mainly on the accuracy ofidentification of the parameters and .

2) The modification for given by (11) holds only ifi.e., steady state is not attained.

When steady state is actually achieved, the value ofshowsa further decrease as time increases. This behavior is due tostatic friction, which becomes significant again after the system

attains a steady state and remains in that position for a while.In other words, the longer the system stays at one steady-statevalue, the harder it is for the system to change the state. Itwas found by experiments that for a fixed input the steadystate is not attained for a step width of less than 5 s, hence inthis case the approximation given by (11) holds true. This isnot a severe limitation considering the fact that actual brakingcommands do fall into this category.

V. CONTROL DESIGN

The nonlinearity of the brake subsystem under considera-tion is in the form of hysteresis and dead zone. The mainobjective of the controller design in this paper is to makethe performance of the brake subsystem as uniform androbust as possible throughout the range of operation. Oneway to achieve this objective is to use feedback linearizationto cancel the nonlinearities of the system. Furthermore, wewill augment the feedback linearized system with suitablydesigned compensator to meet the performance requirementof no overshoot and zero steady-state error for step inputs.



Fig. 16. Comparison of closed-loop response for PI compensator with and without modifications.

The design of the brake controller in this paper follows theguidelines provided by the feedback linearization techniquesgiven in [6]. The brake model given in (15) has no explicitcontrol input term. Since the parameterin the brake modelis a nonlinear function of the control input and state andthe inverse mapping

(16)

is guaranteed to exist for all values of and within theoperating range of the system, where

exists and (17)

where is the maximum allowable pressure andand represent the minimum and the maximum allowablecontrol input, respectively. Hence with the condition given in(17), we can consider to be the virtual control input. Thisassumption would help us to linearize the system by usingstandard input–output feedback linearization techniques. Thecontroller design proceeds by first linearizing the brake model

(15), with output and input , without changing theinternal state dynamics given in (15).

Since we can let

(18)

where is some design constant andis the new input.The resulting system is

(19)

Since given by (18), for some constant and , is asignal to be computed, the control inputcan be calculatedusing the inverse mapping (16). The feedback linearizedsystem is given by (19), whose transfer function is

(20)

The input can now be selected to meet the control objectivefor the feedback linearized system (19). In general, as shownin Fig. 10, could be of the form



Fig. 17. Simulation and actual system response for step input, corresponding to a desired pressure of 200 psi.

where will be chosen to meet the control objective. Forexample, with

(21)

we have a PI compensator whose parameters canbe chosen for stability and zero steady-state error. The looptransfer function with the addition of the PI compensatorbecomes

(22)

Hence with the addition of a PI compensator the order ofthe closed-loop system has increased. This, however, can beavoided by carefully selecting the gains and . Onesuch combination is given by

where is a design constant to place the closed-loop poleat a desired location. Hence the inputin (19) becomes

(23)

The closed-loop transfer function shown in Fig. 10 isgiven by

(24)

The control law is summarized in Table I and is representedby the block diagram shown in Fig. 11.

A. Modifications for Robustness andPerformance Enhancement

The model of the brake system derived in this paper is basedon parameter identification from input–output data. Theseparameters can have significant amount of drift around thenominal values. Despite modeling inaccuracies and parameterdrift, the controller design based on any model should besufficiently robust to guarantee the desired performance whenimplemented on the actual system. In our case, to make theclosed-loop performance more robust with respect to modelingerrors, some additional logic is embedded within the standardPI compensator. This logic along with some justification ispresented below.

• The control input generated by the feedback linearizationgiven in Table I is calculated based on the assumptionthat there is no saturation of the control input. However,



(a)

(b)

Fig. 18. Closed-loop system response for (a) sinusoidal input of 0.25 Hz and (b) 0.5 Hz. The desired pressure is step input of 100 psi from 0 to 10 s, andfrom 10 to 20 s a sinusoidal input with amplitude of 25 psi is superimposed on the step input. It should be noted that system bandwidth is around 0.5 Hz.

from the safety point of view a limited control authorityis available in the given system. The values ofas calculated in (18) can be arbitrarily large duringtransients; however, the inverse mapping,is guaranteed to exist when the inputs are within thephysical constraints imposed by the system (17). Henceto avoid invertiblity problems a saturation functionis introduced at the output of

ififelse

(25)

where and is the maximumallowable pressure defined in (17)

• To avoid the integration wind up problems caused bymodeling inaccuracies a limited integrator, in place of theideal integrator, is used. The limit of integration dependson the values chosen for the controller gains and can beshown to be equal to .

• To avoid possible overshoots caused by ignored dead zonein the system, the logic shown in Fig. 12 is added to the PIcompensator. The branch labeled “delay_kill” in Fig. 12

stops the integrator accumulation during the dead zone.This, in effect, reduces the saturation of the control inputduring this interval and helps to avoid overshoot at lowoperating pressures.

• The PI compensator (23) cancels the pole of the feedbacklinearized system with the zero. This cancellation is notperfect due to approximations involved in the model-ing process, i.e., the feedback linearized system in notexactly the same as represented in (20). This impropercancellation will increase the order of the closed-loopsystem and may result in an overshoot. To avoid thisundesirable effect, the branch labeled “Overshoot_kill”cuts off integrator when either:

1) the nonlinear function used as linearization input,exceeds the maximum steady-state pressure value andthe actual pressure is less than the desired one;

2) when is negative and the brake pressure is greaterthan the desired.

This modification in PI compensator, described above, re-sults in minimum possible overshoot. Hence during dead zoneand at the time when the controller output is saturated, this



Fig. 19. Comparison of the feedback linearized system response with the equivalent linear system response for two different inputs.

logic avoids excessive integrator accumulation, that may causesubsequent overshoots. The output of the PI compensator withthis modification is given as

if

else(26)

where or andor and is

the minimum line pressure in the idle state, when the timedelay is large.

Some simulation results before and after addition of thislogic are shown in Fig. 16. A comparison of the attainableperformance using this modification, with the standard PIcompensator is given in Table II. With the proposed modi-fication, the main characteristics of the closed-loop system aresummarized as follows.

• The step response is fast enough to meet the AVCSperformance specifications [7].

• There is no overshoot or undershoot.• The steady-state error for step inputs is equal to zero.

VI. STABILITY ANALYSIS

We first discuss the stability properties of the open-loopsystem. The nonlinear functions and in (15)

are guaranteed to be bounded, i.e.,

(27)

(28)

Hence for any bounded initial condition, any bounded inputto the system would result in a bounded state, i.e.,

Furthermore, where and are asdefined in (17). As discussed before, the nonlinear function

is used in place of the control input for linearization ofthe open-loop plant (15). As given by condition (17), if weguarantee that then it implies thatwhere forms the set of acceptable inputs to the system. Thefollowing theorem establishes the stability of the closed-loopsystem.

Theorem 1: If the controller given in Table I with themodifications (26), (25) is initialized such thatthen the closed-loop signals and are bounded and

Proof: The assumption that at is notrestrictive as the controller would be initialized with someacceptable input. To establish the boundedness of all thesignals in closed loop, we begin as follows.



We have that Considering thePI compensator, we can write

(29)

(30)

where is the desired output trajectory, obviously2 Since is a stable

proper transfer function, hence forwe have Now from (19) we have that

Similarly from (26) we have thatDenoting in (18) by

we have

(31)

where is a finite design constant chosen earlier,Also from (18) we can write

(32)

where and are finiteconstants asFinally the saturation function defined in (25) ensuresthat

Since, from (17) we getNow by induction we can show that closed-

loop signals are bounded and for all

VII. SIMULATION AND IMPLEMENTATION RESULTS

The modified control law given in (16), (18), (26), and (25)is simulated using Matrixx and the nonlinear brake model. Theblock diagram of the closed-loop system is shown in Fig. 13.The simulation results of typical braking scenarios are shownin Figs. 17 and 18. The simulation results confirm the claimsmade about performance in terms of zero steady-state error,no overshoot, and sufficiently fast response (limited by thesystem constraints).

The controller given in Fig. 13 is also implemented onthe actual brake system. The results obtained from the ac-tual closed-loop system are shown in Figs. 17 and 18. Thesimulation results in Fig. 17 are almost identical to the actualclosed-loop system response shown in Fig. 18. Furthermore,one of the design objectives to make the system behavelinearly is proved by comparing the actual closed-loop systemresponse with that of the equivalent linear system

This comparison for two different inputs is shownin Fig. 19.

VIII. C ONCLUSION

A nonlinear model that describes the input output behaviorof the brake subsystem of a Lincoln town car is developed.The model is simplified to resemble a first-order linear system

2A function f(x) is O(x) if there exists a finite constantc> 0; such thatjf(x)j � cjxj 8x:

TABLE IIILEAST-SQUARE FIT VALUES OF FUNCTIONS g,h, AND g� FOR THE CURVES IN FIGS. 6 AND 7

with nonlinear coefficients. The unknown parameters are iden-tified by applying the standard curve-fitting techniques to thedata obtained by conducting experiments on the test bench.The hysteresis phenomenon is modeled by isolating the twooperating modes, that is the building and bleeding modes, andidentifying separate sets of parameters for each one. The effectof dead zones and friction on the system output has beenquantitatively modeled as a percentage change in the systemtime constant . The worst case modeling error was foundto be less than 5% within the range of interest.

A brake controller is developed using standard feedbacklinearization techniques on the nonlinear validated model. API compensator with intuitive modifications is introduced inthe closed loop to meet the performance specifications. Thecontroller has been shown to meet the desired performance



TABLE IVLEAST-SQUARE FIT VALUES OF b = h�(u; x) FOR THE CURVES IN FIG. 7. AN X AS A TABLE ENTRY INDICATES AN INVALID STATE FOR BLEEDING MODE

requirements in an AVCS application by applying to the actualbrake subsystem mounted on a bench.

APPENDIX ILOOKUP TABLES

See Tables III and IV.

REFERENCES

[1] J. C. Gerdes, D. B. Maciuca, P. E. Devlin, and J. K. Hedrick, “Brakesystem modeling for IVHS longitudinal control,”Advances in RobustNonlinear Contr. Syst., vol. 43, ASME, Nov. 1993.

[2] Y. Khan, P. Kulkarni, and K. Youcef-Toumi, “Modeling, experimen-tation, and simulation of a brake apply system,” inProc. 1992 Amer.Contr. Conf., 1992, pp. 226–230.

[3] P. Ioannou and Z. Xu, “Throttle and brake control system for automaticvehicle following,” IVHS J., vol. 1, no. 4, pp. 345–377, 1994.

[4] A. S. Hauksdottir and R. E. Fenton, “On the design of a vehiclelongitudinal controller,” IEEE Trans. Veh. Technol., vol. VT-34, pp.182–187, Nov. 1985.

[5] M. J. Nunney,Light and Heavy Vehicle Technology, 2nd ed. Oxford,U.K.: Oxford Univ. Press, 1992, pp. 516–552.

[6] J. E. Slotine and W. Li,Applied Nonlinear Control. Englewood Cliffs,NJ: Prentice-Hall, 1991.

[7] S. E. Shladoveret al., “Automatic vehicle control developments in thePATH program,”IEEE Trans. Veh. Technol., vol. 40, pp. 114–130, Feb.1991.

[8] S. E. Shladover, “Longitudinal control of automotive vehicles in close-formation platoons,”ASME J. Dynamic Syst., Measurement, Contr., vol.113, pp. 231–241, 1991.

[9] S. Sheikholelslam and C. A. Desoer, “Longitudinal control of a platoonof vehicles,” in Proc. 1990 ACC, May 1990.

[10] P. Varaiya, “Smart cars on smart roads: Problems of control,”IEEETrans. Automat. Contr., vol. 38, pp. 195–207, Feb. 1993.

[11] C. C. Chien and P. Ioannou, “Automatic vehicle following,” inProc.1992 Amer. Contr. Conf., Chicago, IL, June 1992, pp. 1748–1752.

[12] J. K. Hedrick, D. McMahon, V. Narendran, and D. Swaroop, “Longi-tudinal vehicle controller design for IVHS system,” inProc. ACC, vol.3, June 1991, pp. 3107–3112.



Humair Raza received the B.S. degree in electricalengineering from the University of Engineering andTechnology, Lahore, Pakistan. He received the M.S.degree in electrical engineering from the Universityof Southern California, Los Angeles, where he iscurrently a Ph.D. candidate.

His research interests include adaptive control,fuzzy control, and modeling and control of advancedvehicle systems.

Zhigang Xu (M’94) received the B.S. degree in1982 from Jiangsu Institute of Technology, Zhen-jiang, China, the M.S. degree in 1984 from EastChina Institute of Technology, Nannnnjing, China,and the Ph.D. degree in 1991 from the Universityof Texas at Dallas.

He worked as a Postdoctoral Fellow from 1991to 1992 and a Research Assistant Professor fromDecember 1992 to June 1995 in the Department ofElectrical Engineering and Systems of the Univer-sity of Southern California, Los Angeles, and as

a Senior Controls Engineer with MetroLaser Inc., Albuquerque, NM, fromJune 1995 to March 1996. He is now a Software Engineer with NorthernTelecom, Richardson, TX. His research interests include nonlinear control,IVHS, pointing and tracking control.

Bingen Yang received the B.S. degree in Engineer-ing Mechanics from Dalian Institute of Technology,China, in 1982, the M.S. degree in applied mechan-ics from Michigan State University, East Lansing, in1985, and the Ph.D. degree in mechanical engineer-ing from the University of California at Berkeley in1989.

He is currently an Associate Professor of Me-chanical Engineering at the University of SouthernCalifornia, Los Angeles. He is an author of morethan 60 technical publications. His research interests

include vibration and control of complex mechanical systems, modeling andcontrol of distributed parameter systems, dynamics of combined flexible-rigidbody systems and gyroscopic systems, vibration isolation of structures, smartstructures, and advanced highway/vehicle systems.

Dr. Yang is Associate Editor of theASME Journal of Vibration andAcoustics.

Petros A. Ioannou (S’80–M’83–SM’89–F’94) re-ceived the B.Sc. degree with First Class Honorsfrom University College, London, U.K., in 1978 andthe M.S. and Ph.D. degrees from the University ofIllinois, Urbana, in 1980 and 1982, respectively.

In 1982, he joined the Department of Electri-cal Engineering-Systems, University of SouthernCalifornia, Los Angeles, where he is currently a Pro-fessor and the Director of the Center of AdvancedTransportation Technologies. His research interestsinclude adaptive control and applications, intelligent

transportation systems, vehicle dynamics and control, and neural networks.


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