Active Airbag Systems Modelling and feedback control design Master's thesis DCT nr.: 2004.79
H. J.L.M. Consten
Engineering Thesis Committee: Chairman: Prof.dr.ir. M. Steinbuch Coach: Ir. R.J. Hesseling Dr.ir F.E. Veldpaus Prof.dr.ir. P.P.J. van den Bosch
Eindhoven, July 2004
Abstract Nowadays, most passenger cars are equipped with airbags. The purpose of the airbag is to prevent a collision between the occupant and the interior of the car during the crash by absorbing its kinetic energy. A major disadvantage of state of the art airbags is that they are not adjustable to crash and occupant characteristics. In this thesis, a feedback control approach is proposed to manipulate the inflator and vent size of the airbag such that the head acceleration follows an a priori defined reference signal. The head acceleration is used because the maximum acceleration appropriately represents the sustained injury. Preferably, bag pressure is to be manipulated. However, that is not possible and therefore, mass flow and vent size are used. An inflator or gas-generator supplies gas, increas ing the pressure inside the airbag. A vent of certain size enables gas to flow out, reducing the pressure. A model of the vehicle and occupant subjected to a crash is available but not suitable for control design. The crash model is a very complex and nonlinear model and therefore not suited for controller design. Therefore, LTI models are derived that can be used for controller design. The relevant transfers from inflator to head acceleration and from vent size to head acceleration are modelled, using the approximate realization method. Loop shaping is used to design controllers. From the chosen injury criteria, requirements for the reference signal are derived. These requirements combined, yield a reference signal that reduces the risk of injury significantly with respect to the original crash. Controllers that satisfy both stability and performance criteria are designed. The controllers of both inputs are implemented separately and the drawbacks of using one input become clear. As already mentioned, the main problem is that the inflator can only increase the pressure and the vent-size decrease the pressure resulting in either a higher or lower deceleration, respectively. Several possible implementations of both controllers simultaneously, are evaluated. The method based on a switch that uses the controller outputs to decide what input to activate, yields the best results.
Abstract (Dutch) Tegenwoordig zijn de meeste personen auto's uitgerust met airbags. Het doe1 van de airbag is het voorkomen van een botsing van de inzittende met het interieur van de auto tijdens een crash door de kinetische energie te absorberen. Een groot nadeel van state of the art airbags is dat ze zich niet kunnen aanpassen aan eigenschappen van de crash of inzittende. Daarom is er een feedback control aanpak voorgesteld om de inflator en gat grootte van de airbag t e regelen zodat de hoofd versnelling een a priori gedefinieerd referentie signaal volgt. De hoofd versnelling is gekozen omdat de maximale acceleratie het opgelopen letsel goed representeert. De airbag heeft dus twee inputs die gebruikt kunnen worden om de hoofd versnelling een referentie signaal te laten volgen. Een inflator of gas-generator levert gas om de druk in de airbag te verhogen. Een gat van bepaalde grootte geeft de mogelijkheid om gas t e laten ontsnappen om zo de druk t e verlagen. Er is een model van het voertuig en inzittende tijdens een crash beschikbaar maar dat is niet geschikt voor regelaar ontwerp. Het crash model is erg complex en niet-lineair en daarom niet geschikt voor controller design. Daarom zijn er LTI modellen gemaakt die we1 gebruikt kunnen worden voor regelaar ontwerp. De relevante overdracht van ingaande massa stroom naar hoofd versnelling en van gat grootte naar hoofd versnelling is gemodelleerd, gebruik makend van de approximate realization methode. Loop shaping is gebruikt om regelaars te ontwerpen. Uit de gekozen letsel criteria volgen eisen waaraan het referentie signaal moet voldoen. Een combinatie van deze eisen levert een referentie signaal dat de kans op letsel verkleint vergeleken met de originele crash. Er zijn regelaars ontworpen die aan de gestelde stabiliteits en prestatie eisen voldoet. De regelaars van beide ingangen zijn een voor een geimplementeerd en zo worden de nadelen van het gebruik van slechts een ingang duidelijk. Zoals a1 eerder vermeld, is het grootste probleem dat de inflator de druk alleen kan verhogen en de gat grootte de druk alleen kan verlagen hetgeen resulteert in respectievelijk een grotere en lagere deceleratie. Enkele mogelijke implementaties om beide controllers tegelijkertijd toe t e passen zijn geevalueerd. De methode die gebaseerd is op een schakeling die de uitgang van de regelaars gebruikt om te beslissen welke ingang geactiveerd wordt, levert het beste resultaat OP.
Contents Abstract
1
Abstract (Dutch)
iii
Introduction
1
1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 State of the art passive safety systems . . . . . . . . . . . . . . . . . . . .
2
................................ 1.2.2 Airbag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Disadvantages of state of the art airbag and belt . . . . . . . . . .
2
1.2.1
Seat belt
1.3 Automotive crash testing
3
4
........................... 4
1.4 Injurycriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.5 The numerical simulation model . . . . . . . . . . . . . . . . . . . . . . .
6
1.6 Feedback control of airbag systems . . . . . . . . . . . . . . . . . . . . . .
8
1.7 Objectives and outline of this report . . . . . . . . . . . . . . . . . . . . . 10 2 Modelling for airbag control
11
2.1 Modelling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2
Minimal state-space realization in linear system theory . . . . . . . . . . .
13
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2.1
. . . . . . . . . . . . . . . . . . . . . . 13 2.2.3 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Approach applied to the crash model . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Time instants to add step perturbations . . . . . . . . . . . . . . . 17 2.3.2 Size of the step perturbation . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 System identification of the transfer from to xh . . . . . . . . . 21 2.3.4 System identification of the transfer from A, to xh . . . . . . . . . 23 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2
Minimal realization theory
CONTENTS 3 Controller Design 3.1 Reference Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2
Controller Setup and Design Criteria . . . . . . . . . . . . . . . . . . . . .
3.3 Controller design and implementation . . . . . . . . . . . . . . . . . . . .
27 28 30 31
4i,controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
A, controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Evaluation of the two inputs . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Identification along closed loop trajectory . . . . . . . . . . . . . . . . . . 3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.3.1 3.3.2
4 Combined manipulation of inputs
37 39 40 41
4.1
Straight forward combination of both inputs . . . . . . . . . . . . . . . . . 42
4.2
Error based switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Controller-output based switching . . . . . . . . . . . . . . . . . . . . . . 46
4.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
45
Conclusions and Recommendations
51
Bibliography
53
A Airbag and belt system components
55
A.l Airbag components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A.2 Seat belt systems components . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.2.1 load limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.2.2 Pretentioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A.2.3 Retractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
B Additional figures for Chapter 2
59
C Additional figures for Chapter 3
63
D Additional figures for Chapter 4
65
E Symbols
67
Chapter 1
Introduction 1.1 General introduction Over the last years, vehicle safety has become very important. Both the government as well as the automotive industry put a lot of effort in research, to improve safety. One of the reasons for that is the number of casualties and sever injuries as a result of crashes. To get an impression of the importance of this subject, some figures considering road safety in the EU in the year 2001 are given:
Approzimately 97 percent of all transport deaths in the EU are related to road crashes. Road crashes are the leading cause of death and hospital admission for EU citizens under 50 years. Road accidents lead t o more than 93 percent of all transport crash costs. They cost more than congestion and pollution, or cancer and heart disease. I n 2001, the total of road fatalities in the EU was 38,935. [Source: European Transport Safety Council ([I ti])] Road statistics reveal two predominant causes for fatal road crashes: inappropriate speed and alcohol. Inappropriate speed does not necessarily mean that a speed limit is violated. It can also mean that the speed is not adjusted to the circumstances. Research has shown that alcohol reduces the capability of a person to handle a car. The best solution is to separate alcohol and traffic completely but since alcohol is an important part of our culture, a small amount is allowed when driving a car. The measures to reduce the number of casualties and injuries can be divided into two groups P21, 8
Active safety measures to avoid a crash. For example, driver fitness, vehicle maintenance and road signalling.
0
Passive safety measures to minimize the consequences of an crash. For example, restraint system and crumple zone.
Two possible strategies exist to reduce the number of casualties and injuries. On one side, the responsibility of the driver to drive safely is appealed and on the other side
CHAPTER 1. INTRODUCTION the vehicle is equipped with safety systems. In this report, it is assumed that a crash is unavoidable for whatever reason. Therefore, we focus on passive vehicle safety. More specifically, the safety restraint system consisting of airbag and seat belt. One of the first measures to improve passive safety of a vehicle, was the introduction of the seat belt. Its purpose is to keep the occupant inside the vehicle and makes sure the occupant does not hit the interior of the car. The safety belt already meant a great improvement but some people who survived due to it, still suffered severe head injuries as they bumped into the steering wheel or dashboard. The airbag was developed to prevent a collision between the occupant's head and the interior of the car. At First, airbags were only used for frontal crashes. Research shows that 59% of all vehicle occupant fatalities occur in vehicles that sustained frontal crashes [18]. The other 41% are rollover, side impact and rear impact. The airbag proved to be very effective, reducing the head injury significantly during a frontal crash. Currently, airbags are not only used for frontal crashes but also for lateral collisions. At this moment, cars can be equipped with up to nine airbags.
1.2
State of the art passive safety systems
In this section, a brief overview of state of the art airbags and seat belts is given. The working principles of the various parts of the seat belt and the frontal airbag are explained. Since the contents of this report focusses on the airbag, the seat belt is discussed briefly, merely for completeness, because the occupant is assumed to be wearing a seat belt during all considered crash simulations. More information about both the seat belt and airbag can be found in Appendix A.l
1.2.1
Seat belt
The components of a modern seat belt system are the belt itself, a retractor, a pretentioner, sensors, an electronic control unit (ECU) and a load limiter [6], [7]. A seat belt integrated in a typical safety restraint system is displayed in Figure 1.1. The belt is connected to a retractor mechanism. The central element in the retractor is a spool, attached to one end of the belt. Inside the retractor, a rotational spring applies torque to the spool. This rotates the spool so it winds up any loose belt. The retractor has a locking mechanism that stops the spool from rotating when the car is involved in a collision.
A pretensioner removes any slack in the belt in the event of a crash. Whereas the conventional locking mechanism in a retractor keeps the belt from extending any further, the pretensioner actually pulls in on the belt. This force can sometimes help a little to move the occupant into a better crash position in his or her seat. Pretensioners normally operate in combination with conventional locking mechanisms, not replacing them. Some pretensioners are built around electric motors but the most popular designs today use pyrotechnics to pull in the belt webbing. When one of the sensors sends a crash signal to the ECU, the pretentioner is triggered and the retractor locks. If the crash is very severe due to high speed, the belt may be the
1.2. STATE O F THE ART PASSIVE SAFETY SYSTEMS
sensors ECU
Figure 1.1: Modern automotive safety restraint system cause of injury. To prevent this, a load limiter limits the belt force by plastic deformation of its components. As a result, the belt can be pulled out. Allowing the occupant to move forward and keeping the belt force at a controlled and pre-defined level.
1.2.2
Airbag
An airbag system consists of three components. The airbag modules, crash sensors and the electronic control unit (ECU) [7],[2].The airbag module consists of the bag itself and an inflator. A airbag system integrated in a modern safety restraint system is displayed in Figure 1.1. When the ECU detects a crash signal from a sensor, it triggers the inflator of one or more airbag modules and the bag deploys. The sensor can be an accelerometer and the ECU detects a crash when the deceleration of the vehicle exceeds a certain limit but there are many more possibilities [2]. The bag is made of a thin nylon fabric. It is folded in a specific way and mounted in a cover on the steering wheel or in the dashboard, seat, door or roof. The most common inflators use solid propellants, while hybrid inflators use a combination of compressed gas and a solid fuel. The chemical reaction that starts when the inflator is triggered by the ECU produces a large volume of hot nitrogen gas in a very short time (about 30 ms). It can be compared with a small explosion. The bag bursts through its cover and unfolds at a speed of up to 300 kph. When the airbag is being inflated, it immediately starts to exhaust gas. The airbag fabric is porous, but the majority of the gas flows out of the airbag through a hole of a certain dimension, the so called vent. The deflating bag ensures that the airbag gives way when the occupant's head pushes against it. As a result, the head deceleration is distributed more evenly over the available crash time, resulting in a lower maximum deceleration of the occupant's head. A decrease of the maximum deceleration reduces the risk of injuries for the occupants in the vehicle. The deceleration of the head as a result of the airbag depends on the force on the head generated by the airbag. This restraint force depends on the pressure inside the airbag, pb, influenced by adding gas to or subtracting gas from the airbag. As a result, two inputs to manipulate the pressure can be defined: an inflator to increase the pressure and a vent to decrease the pressure.
CHAPTER 1. INTRODUCTION As can be seen in Figure 1.1, there is a difference between the driver and passenger side airbag. The circular shaped driver side airbag is much smaller and inflates more rapidly than the rectangular shaped passenger side unit. The driver side airbag has less distance to travel before contact with an occupant, since the bag is mounted closer t o the driver via the steering column. As a result, the passenger bag is typically three to five times bigger. Only the driver side airbag is considered in this report.
1.2.3 Disadvantages of state of the art airbag and belt Induced injuries Airbags can also be the cause of injury [8]. By reducing critical injuries, other minor injuries like laceration, abrasion and burn wounds are induced. Airbag induced injuries mostly occur during deployment of the bag, if the occupant is too close t o the steering wheel or dashboard. Limited flexibility A disadvantage is that the state of the art airbag and belt are not really flexible to adjust to the crash circumstances. The parameters of the airbag and belt are tuned to perform well for various crash and occupant characteristics. Because crash circumstances vary a lot, the restraint system sometimes "restraints" too much or not enough. In [I],possibilities to adjust restraint systems during severe crashes are discussed. Examples are a load limiter of a belt, discussed earlier, and a pressure limiter of an airbag. Both apply the same principle: limiting a variable to a certain value. This can be considered as a step towards adjustable restraint systems because their behavior is different during severe crashes than during minor crashes. But still, these systems are not really flexible because the limits are determined during the development of the car, not during a crash. For the belt and bag, the triggering of the retractor and inflator respectively, are the only parameters that depend on the type of crash. In this report, it is examined if an adjustable vent-size and inflator can be used to make an airbag more adjustable to the crash circumstances and to improve its performance. Feedback control is a possible solution to manipulate the inputs of the airbag in such a way that, despite various circumstances, the occupant is provided with the best possible protection for the head, in any type of crash. In [4], feedback control of a seat belt showed encouraging results.
1.3
Automotive crash testing
Since 1978, the United States New Car Assessment Program (US-NCAP) provides consumers with a measure of the relative safety potential of vehicles in crashes. In 1997, Euro-NCAP was founded in Europe. NCAP now supplies consumers with important information about frontal- and side-crash test results. It can help them in their vehicle purchase decisions. The ultimate goal of NCAP is to improve occupant safety by providing market incentives for vehicle manufacturers to voluntarily make their vehicles safer.
1.4. IAJJURY CRITERIA
Car manufacturers perform crash-tests to determine the safety of a car during its design process, but in order to insure the objectivity of the tests and to compare different car brands, independent organizations like US-NCAP and Euro-NCAP were founded. Both organizations subject a new car to various tests like a frontal crash, a side crash and a roll-over crash. Dummies with sensors to measure the sustained injuries due t o a certain crash, are used to simulate the occupants. In this report, the standard US-NCAP frontal crash test is used to evaluate airbag manipulation strategies. As already mentioned, the majority of all crashes are frontal crashes. The US-NCAP crash test simulates a collision of a car hitting a fixed rigid barrier with a forward speed of 56 kph.
1.4
Injury criteria
A rating system was introduced by the NHTSA' t o be able to compare crashes in an easy way. The method consists of injury criteria for several parts of the body. In frontal crashes, the rating is determined by the worst score on injury criteria: Head Injury Criteria e
Chest deceleration
a
Femur load
The head injury criteria are used to determine the performance of an airbag during this project. The injury t o the head is considered since the airbag was initially designed to reduce the risk of head injury. Objectives of using an airbag are, prevent the head to touch the interior, increase deceleration distance of the head and distribution of the contact force over a larger surface. From the numerous injury criteria, the following are used in this report, e
Intrusions into the body of the dummy should be prevented as much as possible. No contact between torso and head of the passenger and the vehicle may occur other than airbag or belt contact.
a
Head Injury Criterion (HIC), proposed by the NHTSA,
where time is t [ms], tl and t2 are two arbitrary points in time during the deceleration pulse of the head, -xh [m/s2]. The deceleration is measured in multiples of the acceleration of gravity, [g] . A time interval, proposed by the NHTSA, of t2- tl = 36 [ms] is used t o determine the HIC value in this report. The standard HIC36 will be denoted as HIC from now on. A low HIC value means a low risk of injury. 'National Highway Traffic Safety Administration ([17])
CHAPTER 1. INTRODUCTION Although other criteria can be defined and other parts of the body can be considered, this report considers the injury criterion HIC to evaluate the risk of injury for a particular crash case. Therefore, a strategy is needed to reduce the HIC value.
1.5
The numerical simulation model
The engineering software tool Madymo [20] is used to simulate a crash and test the performance of new airbag or belt design. In the automotive industry, Madymo is one of the crash test simulation programs that is often used to asses safety systems because real crash tests are time and money consuming. Furthermore, practical limitations with respect to sensors and actuators make it impossible to test the proposed strategy during a real crash test. Madymo is considered t o be a realistic representation of a real crash with no limitations to actuator or sensor performance. In [I41 it is shown that crash test simulations can be a good alternative for real crash tests. The Madymo model consists of a multi-body part (for example seat and dummy) and a finite-element part (for example airbag). A graphical representation of the complete crash model is displayed in Figure 1.2.
(a) Dummy in initial position
(b) ISO-view of the dummy and inner vehicle
Figure 1.2: The numerical crash model of the BMW E46. It is a model of the BMW E46 (3 series) with a 50th percentile male dummy, developed by TNO and BMW. It represents a normal sized family car. The HYBRID I11 Fiftieth Percentile crash test dummy, representing the average adult male, is the most widely used dummy in frontal crash tests. It represents the human dynamics in a realistic way. In the simulation model, this dummy is modelled numerically. The total crash model consists of over 180 rigid bodies attached to each other with springs and dampers and a Finite Element Model of the airbag of over 2850 elements. The crash model is nonlinear and very complex. To simulate a crash, the position in time for the car part of the crash model is prescribed. For the dummy, only an initial velocity is given. It is able to move freely in forward direction and it's deceleration can therefore be
1.5. THE NUMERICAL SIMULATION MODEL
time [ s ]
time [ s ] (b) Original vent size, A,,b(t)
(a) Original ingoing mass flow, &,o(t)
-
0
0.02
0.04
0.06
0.08
0.1
0.12
time [ s ] (c) Original head xh,~(t)
acceleration
occupant,
Figure 1.3: Time history of a crash simulation with state of the art airbag and belt influenced by the airbag and seat belt. For more information on the simulation program Madymo, see [20]. As already mentioned, the airbag has two parameters that can be varied, i.e. two inputs. The ingoing mass flow generated by the inflator and the size of the vent can be defined as a function of time. Normally, the ingoing mass $in and vent size A, are supplied in a time-table. The two signals are adjusted to a particular combination of vehicle and airbag that delivers relatively good performance for different occupants and crash types. The original inputs for and A, are displayed in Figure 1.3(a) and 1.3(b) respectively. din is mass flow rate of a typical inflator triggered at t = 6 m s . A, represents a Input standard vent size of 1.73 - 10V3 rn2 that is opened at t = 24.5 m s because initially, no gas can escape through the vent because the bag is folded. Here, the US-NCAP frontal crash test with the dummy positioned at the driver-side, is considered. The head acceleration in forward direction, xh of the dummy wearing a seat belt during this crash is shown in Figure 1.3(c). The peaks at t = 24 m s are due
CHAPTER 1 . INTRODUCTION
(a) maximum head (64 ms)
deceleration
(b) closest to the steering wheel (78 ms)
Figure 1.4: Kinematics of vehicle and dummy at a certain time during a crash to the unfolding bag, shortly impacting or slapping the head. The deceleration has a peak of 614 m/s2 at t = 64 ms. The HIC value for this crash is 569. The original inputs $ ,, and A, and the original head acceleration will be referred to as Av,o and xh,, respectively.
In Figure 1.4, the kinematics of the original crash are displayed. Figure 1.4(a), shows the position of the dummy at the moment the head deceleration is at its maximum. In Figure 1.4(b), the smallest distance between head and steering wheel can be seen. This distance is relatively large and offers room to enlarge the deceleration distance of the occupant's head. Since the total kinetic energy that needs to be absorbed by the airbag remains the same for a particular crash, a longer deceleration distance results in a lower maximum deceleration. From the earlier defined head injury criteria it can be concluded that the maximum acceleration should be reduced to decrease the risk of injury. Note the fact that in the beginning the two inputs are working against each other. The goal is to get the airbag in a fully inflated state to receive the head as soon as possible with the inflator. But at t = 24.5 ms the vent is uncovered by the unfolding of the bag and the airbag starts to deflate whereas the gas generator is still inflating the airbag. This situation arises since all airbag parameters are set to an average value and cannot be changed during a crash. At a later point in time during the crash, the vent needs to be open to reduce the pressure inside the airbag. Therefore, it is always present although it is not needed for some time periods.
1.6
Feedback control of airbag systems
The occupant and airbag interaction can be considered as a dynamic system. The output y(t) can be manipulated by changing the value of the two inputs of the airbag: ul(t) and up(t) denoting the ingoing mass flow and the vent size. The values of the two inputs can be controlled with a feedback controller C. A system variable is chosen as output y(t), and has to follow a prescribed reference signal. If the output follows this reference
1.6. FEEDBACK CONTROL OF AIRBAG SYSTEMS
Figure 1.5: Schematic representation of the control strategy. The crash model is represented by M , containing the car with airbag and the occupant. C is the controller manipulating the ingoin,g mass flow and vent size of the airbag. Input of the controller is the error between a specific system variable and the reference signal for this variable. signal, the risk of injury is considered to be low. It is assumed that the reference signal is constructed such that the HIC value is considerably lower than for the original crash with state of the art bag and belt. To follow the reference, a controller needs to be designed that adjusts the inputs such that the error e ( t ) between the reference r ( t ) and the output is close to zero. Because C has one input and two outputs, inside C the controller also makes sure there is a correct distribution between the two outputs. This control approach is represented schematically in Figure 1.5. The Madymo crash model is too complex to use for control design due to the large amount of multi-body and finite-element equations. Therefore, a modelling approach is needed to derive models that are suited for control design. To keep the design process as easy and clear as possible, the goal is to derive simple
LTI models that can be used for various control design approaches. Furthermore, it is preferred to use loop shaping for control design because performance and stability can be evaluated in a straight forward way. For online manipulation of the inputs, we need to prescribe the two inputs per simulation time step. For this purpose, a coupling between MatlabISimulink and Madymo is developed by TNO. By means of this coupling it is possible to send data from Matlab to Madymo and vice versa. With Simulink, a simulation model can be designed in Matlab. The crash model that is calculated by Madymo, is represented by a Simulink-biock with inputs and outputs. In Simulink, information can be received about the current state of the Madymo model, such as the acceleration or deflection of parts of the dummy or the pressure and volume of the airbag. This up-to-date information can be used to calculate a new value for and A, or another parameter that is a function of time, and can be passed to Madymo by the coupling.
CHAPTER 1. INTRODUCTION
1.7
Objectives and outline of this report
The main objective is:
Development of a strategy t o use controller(s) for the airbag, t o force the head accelemtion t o follow a reference signal. The research goais are, 1. Modelling of the relevant dynamics 2. Reference signal design
3. Control design This comes down to the designing of a correct structure of the controller in block C of Figure 1.5 that translates the error between desired output and actual output into an appropriate signal for the two inputs of the airbag. In Chapter 2, several possible ways to model the dynamics are discussed and the most suitable method is chosen, explained and applied to the crash model. Loop shaping is used to derive a controller for both inputs, in Chapter 3. These controllers are implemented one at a time to verify their performance and stability, both in frequency and time domain. In Chapter 4, different structures for C in Figure 1.5, to use controllers for both inputs simultaneously, are discussed. It is explained why it is necessary to use two inputs and what problems occur when two inputs are used to control one output. Finally, conclusions and recommendations are presented.
Chapter 2
Modelling for airbag control A complex numerical model of the vehicle and occupant subjected to the considered crash, is available. This model cannot be used because it is too large and too complex and nonlinear, based on a combination of Multi-Body-Dynamics and Finite-Element-Method equations. For the same reasons, model reduction cannot be used. Model reduction algorithms are not able to handle models like the crash model. A suitable technique to model the airbag and head dynamics is presented. In this chapter, a dynamic mathematical model will be derived that is suitable to use for controller design. Only the properties and phenomena that are relevant for the purpose of the model should be included in the model. The head acceleration is chosen as the variable that is controlled. The two manipulated variables of the airbag, 4i, and A,, are used to force the head to follow the a priori defined reference signal. The relevant transfers that need to be modelled for controller design are:
2.1
Modelling techniques
Numerous techniques to derive models exist. They can be divided into two basic approaches [211:
1. Experimental modelling Experimental data, obtained by measuring variables of the system to be modelled, is used to reveal systematic relations between variables. These relations are considered as a model. The model is required to be able to explain and predict the behavior of the studied system. Experimental modelling can be divided into the following four subsequent steps: experiment design, data acquisition, data processing and model validation.
CHAPTER 2. MODELLING FOR AIRBAG CONTROL 2. Theoretical modelling Using accepted theory of the underlying sciences, equations are derived that describe the basic behavior of a particular system. In general, the equations involved will be conservation laws or other relations that have been derived from first principles. Additionally, the physical properties and other relevant descriptions of the materials in the system need to be described. The complexity and nonlinearity of physical phenomena make it difficult to use theoretical relations for the airbag and head dynamics modelling. Problems occur due to the fact that there is a big difference in dynamics before and after airbag to head contact. After head contact, parameters like the contact area between head and airbag must be determined. Relations to describe the pressure and temperature in the airbag and the volume of the airbag are very complex. Furthermore, a lot of assumptions are necessary to determine the values of the parameters in the equations. The quality of the resulting model is questionable and therefore, they cannot be used for controller design. The experimental modelling approach is more attractive. An advantage of this method is that only relevant dynamics between a chosen input and output are approximated. Therefore, the resulting model is a good representation of the specific part of interest of the system with, in general, a small amount of equations. Numerous experimental approaches are available but due to the numerical crash model, a lot of them cannot be applied. Preferably, the method should apply to the following criteria: e
A step response is preferable as input because numerical models can handle steps better than noise or impulse inputs. It is an advantage if no a priori knowledge of the system is required.
e
Computational noise is always a phenomenon that needs to be dealt with when experimental modelling is applied. An algoritme is needed that does not suffer from noisy data since Madymo produces computational noise.
A method that yields LTI models is preferred that can be used for a great range of control design techniques.
A well known method is the approximate minimal state-space realization method [lo]. This method is chosen because it applied to all criteria. Furthermore, approximate realization has proven itself for the seat belt in this simulation model [4]. Data based approximation will be used to linearize the crash model around a number of operating points along the original head acceleration (Figure 1.3). The reason to use a number of operating points is that the crash model is too complex to obtain a model of the full dynamics. Furthermore, time dependency of the dynamics can be investigated by comparing the responses and state-space models of the different operating points. The linearity of the system dynamics for small perturbation steps around the operating point is investigated by using various step sizes for every operating point.
2.2. MINIMAL STATE-SPACE REALIZATION IN LINEAR SYSTEM THEORY
2.2
Minimal stat e-space reaPizat ion in h e a r system theory
In this section the method to derive a minimal state-space realization of the complex crash model is presented. This method is explained in detail in [lo] and summarized in this section because basic knowledge of the technique is considered important. This strategy is based on Input/Output (110) data of a dynamical system. Minimal statespace realization is based on impulse response data which is in a sequence of Markov parameters but also step response data can be used. The basic impulse response approach is discussed before the modification is presented to use step response data. It is also explained why step response data is preferred to impulse response data. The objective is to derive the system matrices (A, B, C, D ) of Equations 2.4 and 2.5.
2.2.1
Notation
A continuous-time Linear Time Invariant (LTI) system with m inputs, I outputs and n states in state-space form is given by:
+
y (t) = Cx ( t ) Du(t).
(2.3)
and D E EtLxrn.The input of the system is denoted with A E Rnxn,B E Rnxm,C E by u, y the output, x the state and t time. A standard discrete-time LTI system can be formulated as: x(k 1) = Ax(k) Bu(k), (2.4)
+
+
If A is a matrix than aij and A ( i , j ) denote the component of A on the i-th row and the j-th column. For example A(:, k : I) denotes all rows and columns k to I (k, k 1,k 2,. . . ,1). A, is an m x n matrix.
+
2.2.2
+
Minimal realization theory
From the discrete-time state-space system description in Equations 2.4 and 2.5 it is clear that the output of the LTI system to an impulse input at k = 0, is:
This sampled impulse response is equal to the sequence of Markov parameters, {Gk)EO,of the LTI system,
G
From Equation 2.7, matrix D can be obtained directly from the Markov parameters.
=
CHAPTER 2. MODELLING FOR AIRBAG CONTROL From the impulse response data, a Hankel matrix is constructed in the following way,
and the shifted Eaakel matrix
H N (G)
is defined as:
Silverman [13] states that an infinite sequence of Markov parameters, G can be realized if positive integers r , r' exist, such that,
In this case p is the order of the system, equal to n. Ho's algorithm [5] consists of three steps. First the order, p, of the minimal state-space realization is determined by:
with r large enough to satisfy Equation 2.10. Secondly, nonsingular matrices P and Q need to be determined to yield
A particular decomposition to determine P and Q will be given later. Thirdly, the state-space matrices ( A ,B, C, D ) are defined as
with E,,,, the p x q block matrix
[ Ip
1.
Op,g-p The algorithm of Youla and Tissi [15] states that a decomposition like 2.12 can be obtained by: &,,I (G) = HoHc (2.14)
with H, E RTIXpand H, E RpxTm and (rank H, = rank Hc = rank H,,,,(G) = p). In that case it can be proved that
2.2. MINIMAL STATE-SPACE REALIZATION I N LIhrEAR SYSTEM THEORY
20
20
r
4
0
e
10
$5,
3
-20
$?
4 0
8
-60
m
0
Z
%
4
3
-2 r(
Y u
I
-lo
-20
m
'
0
0.01
0.02
0.03
0.04
time [ s ]
-80 -100
0
0.01
0.02
0.03
0.04
time [ s ]
Figure 2.1: Response of the crash model t o impulse and step inputs for compared t o the response generated with a n approximate state-space realization model. with H+ the pseudo-inverse of the matrix H . A good method to determine a full rank decomposition of a matrix and to construct the pseudo-inverse is the singular value decomposition (SVD). The SVD of a matrix H,,,! (G) yields H,,,I ( G ) = U C V ~ Finally, . Ho and Hc are defined as
The system matrices A, B and C can be determined using Equation 2.15
2.2.3
Practice
In practice, the measured response is never equal to the Markov parameters due to the influence of noise. This usually results in the fact that the Hankel matrix H,,,!(G) of Equation 2.14, is of full rank such that p cannot be determined as presented in Equation 2.11. Instead, the SVD of the Hankel matrix can be used. Normally, the SVD results in a matrix C with p non-zero terms on the diagonal. Now there will be p significant singular values before their magnitude drops significantly. Therefore, the number of significant singular values needs to be determined to obtain the order of the LTI model. In practice the approach of Section 2.2.2 will be referred to as approximate state-space realization method As mentioned before, the numerical crash model can handle step inputs better than noise or impulse inputs. To show this effect, simulations performed with the crash model with step and impulse inputs are displayed in Figure 2.1. From this figure, it can be concluded that the response of the LTI model is a good approximation of the airbag and head dynamics of the crash model since the error between the acceleration signals are very small for the first 40 m s of the response. However, the response of the crash model disturbed with noise. The impulse response of the crash model has a much smaller signal-to-noise ratio than the step response. Therefore, it can be concluded that a step response is more suitable for data bases approximation than impulse response data. The step response data can be differentiated to obtain an impulse response. This is not a preferable alternative for impulse response data, since this amplifies the high-frequent
CHAPTER 2. MODELLING FOR AIRBAG CONTROL
noise that is, though much less than in the impulse response, also present in the step response. The minimal state-space realization method, can also be used for step response data. The Hankel matrix H,.,, is modified to,
+
+
with r r' = N 1 and {Sk)f=othe measured step response. From now on, the step response based approach is meant when referring to the minimal state-space realization method. In practice, the length of the step response sequence is limited to N in stead of m. Length N, needed for the approximate state-space realization approach to yield good LTI models for this particular project, needs to be determined. Length N must be large enough to satisfy Equation 2.10. Due to the complexity of the crash model, only a small region around the operating point of the Madymo simulated step response data can be approximated with LTI models. Therefore, N must not be too large because the further away from the step time, the less the relevant dynamics that need to be modelled are represented in the step response. During the second part of the crash, nonlinear effects occur which make the linearized model only valid around the operating point at time t = rJ. The car and occupant start moving in opposite direction, the bag pressure is very low and at the end of the crash the acceleration of vehicle and head are zero. The trade off between N that is large enough to satisfy Equation 2.10 and small enough to eliminate nonlinear effect from the response, is not trivial. In the next section arguments are given for the chosen modelling time-horizon N.
2.3
Approach applied to the crash model
The approximate state-space realization method, described in the previous section, will now be used to derive LTI models to approximate the dynamics of the airbag-head interaction Along the original time-history of the head acceleration of a standard USNCAP crash, the response in xh to step perturbations in the inputs of the airbag is examined. The original inputs are g5in,~(t)in Figure 1.3(b) and AVlo(t) in Figure 1.3(c). Steps are added to these original inputs to generate step response data that can be used for the approximate state-space realization method. Only one transfer at a time is examined, therefore only one input is perturbed. In Section 2.3.3, the transfer from to Sxh is modelled,
with +i,,o(t) and A,,o(t) the original inputs, 0 the unit step at t = ri and Ai the
2.3. APPROACH APPLIED TO THE CRASH MODEL
magnitude of the step. In Section 2.3.4, the transfer from 6A, to 6xh is modelled,
The unit step function is,
This results in a response,
with xh,t,t,i the total head acceleration of the dummy for a step size Aiand step time ~ j xh,0 , the original acceleration and it is assumed that xh,ij(t) is the acceleration due - T ~ and ) step size &. to the step input @(t The modified approximate realization method expects step response data from a unit step in the input. Therefore, the original respons is subtracted from the simulation response to get the step response and the size of the step must be taken into account. From Equation 2.21 the normalized step response for the approximate realization algorithm can be obtained using:
Since a number of different step sizes are used to check linearity, the mean normalized step response for one particular step time ~ j is, calculated by,
with n the number of different step perturbations Ai at t = rj. It is important to determine appropriate points in time, 7-j to add steps to the input because the dynamics of the airbag and head can change as a function of time. Another important property is the step size &. The step size must not be too small or too large. If the step is too small, the response is in the order of the noise. If the step is too large the volume and pressure in the airbag change too much relative to it's initial value to result in a linear respons of the head acceleration. The step response of both negative and positive step sizes are examined to verify linearity in both directions (increasing and decreasing input values).
2.3.1
Time instants to add step perturbations
The results of the original crash test (Figure 1.3(a)) show that the crash takes place between 0 and 100 ms. Between 30 and 100 ms the head of the dummy is in contact with the airbag thus between those two boundaries, system identification is possible. At
CHAPTER 2. MODELLING FOR AIRBAG CONTROL 30 m s the head is only just starting to touch the airbag so the contact area is to small to transfer force to the head. At 40 m s there is stable contact between head and airbag which makes it possible to start adding perturbations to the inputs for modelling. The start time for the data based approximation is t = 40 ms. The last time instant that a step input can be added to yield a good response is before 100 ms, the deceleration of the dummy due to the airbag is finished and identification is no longer possible. Since a certain set of step response data is needed, the final time that step perturbations can be added is earlier than t = 100 ms. T'ne value of N milst be large enough to ensure that the relevant number of singular values does not increase when the value of N is increased. Furthermore, effects at the end of the crash are not relevant and disturb the step response. Arbitrarily, the final time to add a step to the input is chosen at t = 80 ms. To check time dependency of the crash model, step perturbations are added at equidistant time intervals of 10 ms between 40 and 80 ms. This results in a linearization of the crash model in the following operating points, T = [4O 50 60 70 80 ] [ms].
2.3.2
Sizeofthe step perturbation
-250
1 0.05
0.07
0.09
time [ s ]
0.1 1
-1500'
I
'
0.05
0.07
0.09
0.11
time [ s ]
time [ s ]
(4 A, Figure 2.2: Injuence ef stepsize o n the identi,fication results.
2.3. APPROACH APPLIED T O THE CRASH MODEL
The step size must be large enough to excite the dynamics but not too large because a large step might cause nonlinear effects to arise more explicitly in the step response. If the step is too small, the signal to noise ratio of the step response is t o small. To investigate all these effects, an evaluation of different scaling factors from Ai = 0.02 to Ai = 0.25 is performed with steps at an arbitrary chosen step time, rj = 50 ms. For steps in +in the results are displayed in Figure 2.2(a) and 2.2(b). For Ai < 0.15 respases are mere er less the same. Fm Ai = 0.20 and Ai = 0.25 respmses starts t e differ from the once obtained using smaller steps, showing nonlinear influences. Therefore, it is concluded that & = 0.15 is the maximum size of the step. The responses of Ai = 0.02 and Ai = 0.04 show a lot more noise then Ai = 0.06 and Ai = 0.1. Therefore, A = 0.06 is the lower boundary. The system identification will be done with the step sizes of A+ = [0.06 0.08 0.10 0.121. The results for different step sizes for A, are shown in Figure 2.2(c) and 2.2(d). The same evaluation as for 4in can be done and the same arguments can be given to use AA, = [0.06 0.08 0.10 0.121. The size of the step for the vent-size a factor of the m2. Only the effect of larger step sizes is not so obvious as original input of 1.73 . for +in but for convenience the identification of the transfer from A, t o xh will be done with the same step sizes as for & , .
time [ s ]
time [ s ]
Figure 2.3: Evaluation of dzflerence between negative and positive step perturbations. W i t h orig. the original inputs +in,o and A,,-,, pos. positive steps for 4i, in (a) and (b) and neg. negative steps for A, in (c) and (dl.
CHAPTER 2. MODELLING FOR AIRBAG CONTROL
time [ s ] (a) step-perturbated inputs for mass-flow
time [ s ] (b) step-perturbated inputs for vent-size A,
$in
Figure 2.4: Visualization of the inputs used t o generate step response data for the approximate state-space realization method
Until now, only positive steps are considered. An evaluation is done with negative steps, A =(-0.06 -0.08 -0.10 -0.121, to determine if the crash model is linear in both directions. Thus if the models derived using the approximate realization method are valid for both increasing and decreasing values of the input. The results of simulations using steps in 4in are displayed in Figure 2.3(a) and 2.3(b) and for A, in Figure 2.3(c) and 2.3(d). The step perturbations for q5in are added at t = 40 ms, because at t = 55ms, the original input is zero and negative mass-flows are not possible for the inflator of the crash model. Aker rj = 40 ms, the effect of negative mass-flow cannot be determined. The step response of a perturbation at 40 ms gives about 10 ms of step response that can be used, depending on the step size. For A,, rj = 50 ms is used, similar as the step size evaluation in Figure 2.2. The normalized step respons are almost identical. It can be concluded that the negative steps also yield a linear response which means that the system is also linear for decreasing inputs. Note that for the purpose of Figure 2.3, only a difference is made between step responses with positive and those with a negative step input. When the step size is accounted for by using Equation 2.22, the different negative and positive responses are equal for A, in Figure 2.3(d). In Figure 2.3(b), this is only true up to t= 50 ms because then the negative step would cause the total input 4i, to become negative. This is physically not possible for the used inflator. In the preceding two sections the time instants for the operating points and the size of the step inputs are determined. They are the same for both inputs q5in and A, except that step-size A is in [ k / s ] for 4in and in [%I for A,. Both rj and A are visualized in Figure 2.4. In this figure, the various step inputs are added to the original input. It can be seen that the steps are relatively small compared to the original inputs. Thus lineair behavior can be expected.
2.3. APPROACH APPLIED T O THE CRASH MODEL
2.3.3
System identification of the transfer from
din to xh
In this section, the results of the system identification of the transfer from din to xh are presented. Simulations are done for all combinations of step sizes and step times defined in the previous section, see Figure 2.4(a). The mean normalized step response is obtained from the head acceleration with Equations 2.22 and 2.23 for every operating point. Like in Figure 2.2, the response of the different step sizes is compared to see if the dynamic behavior is linear. For the chosen step sizes, all normalized responses of the different operating points are close to each other and thus, the crash model can be approximated with linear models. They are depicted in Figure 2.5. In Figure 2.5(b), it can be seen that the step responses of the different operating points indicating that the dynamic behavior is time dependent. The step responses of the different operating points are all shifted to t = 0 to be able to compare them. A difference in slope at the beginning is especially clear for the response at rj = 40 ms. For ~j = 70 m s and rj = 80 m s the peak decreases because the step response is affected by the end of the crash. Although the singular value decompositYon can handle noise, the step response is filtered with a low-pass filter with a cut-off frequency of, f = 1000 H z . To perform Zero-phase forward and reverse digital filtering, a butterworth filter is used. For every operating point, a LTI model is derived using the mean normalized step response. Not the total mean normalized step response data set is used. Because the crash is over at t = 100 m s and a certain margin is used to eliminate the influence because effects due to the end of the crash can be noticed before t = 100 m s . The used time horizon for each operating point: ~j ~ rj
t ~ 0 . 0 9 0 [ sf o] r q = 4 0 , 5 0 , 6 0 m s 5 t 5 0.098 [s] f o r rj = 70 m s ~j 5 t 5 0.097 [s] f o r 7j. = 80 m s
(2.24)
The standard time horizon is rj < t 5 90 m s because at t = 90 m s , the head acceleration and bag pressure are low. The crash is over and therefore it is not possible t o derive
0.04
0.06
0.08
0
0.1
time [ s ]
0.02
0.04
0.06
time [ s ]
(a) Step responses at different operating points
(b) step responses displayed in fig. (a) shifted to t = 0
Figure 2.5: Mean filtered normalized step responses. 21
(T~[ms])
CHAPTER 2. MODELLING FOR AIRBAG CONTROL
6 " "
I
MADYMO resp. LTI model resp.
time [ s ]
Figure 2.6: Singular values and the step response of an approximate state-space realization together with the crash model step response models of the head and airbag interaction. For rj = 70 m s and 7j. = 80 ms, the acquired data set is to small if 2-j< t 5 90 m s . Therefore, the end of the step response used for the approximate realization algorithm is shifted from t, = 90 m s to 95 5 t, 5 100 m s . The actual used intervals t E (70,98) m s and t E (80,97) ms are determined by looking at the quality of the resulting models. The chosen intervals yield the smallest error between the step response of the crash simulation and the corresponding LTI model. The singular values of the Hankel matrix with the step response data, are shown in Figure 2.6 and Table B.1. For the different step times, every time, two singular values are significantly larger than the others, which means the order of the models to describe the dynamics is chosen to be 2. For rj = 50 m s , the step response data of the realized state-space LTI models is displayed together with the crash model step responses in Figure 2.6(b). For completeness, the step responses of the other realized LTI models are shown in Figure B.1. This figure can be used for model validation since the step response of a good model should be able to predict the behavior of the real system. This is obviously the case and it is concluded that the derived models are suitable to use for controller design. The step responses indicate that the dynamic behavior might be time dependent. The LTI model is only able to predict the behavior of the crash model for the first part of the step response. Furthermore the step responses differ for the various operating points. The resulting data of the transfer functions of the different step times is displayed in Table 2.1. The system matrices A, B , C and D are translated into a transfer function notation for second order LTI systems.
2.3. APPROACH APPLIED T O THE CRASH MODEL
k z [Hz]
9.7197.10-~ 2.0751.10V1 2.5511.10-' -5.8434.10~ -5.4070.10~ -5.4430.10' -2.5590.10~ -1.0326.10~ -2.5524.10'
3.8414.10-I 5.1585-lo-' -3.2666-lo2 -4.5040.10~ -2.5222-10' -6.3279
Table 2.1: Model data of the transfer functions from
din
to ith
Figure 2.7: Bode diagrams of the transfer functions from q&,
to xh
with undamped natural frequency w, [Hz], zero z [Hz], relative damping ( static gain k [-I. Bode diagrams of the models is shown in Figure 2.7.
[-I
and
When comparing the LTI models of the different operating points, time dependency seems small because their Bode diagrams are almost the same. Only for operating point rj = 40 ms, the Bode diagram differs from the rest. Because there is no indication that the dynamic behavior is different at the beginning of the airbag and head contact, it is assumed that the modelling process causes the difference. Therefore, the model of rj = 40 ms is considered an outlier and will not be taken into account during the controller design process.
2.3.4
System identification of the transfer from A, to xh
In this section, the results of the system identification of the transfer function of A, to xh are presented. Simulations were done for all combinations of step sizes and step times (Figure 2.4(b)). The same procedure as in the previous section is followed. Like in Figure 2.2, the response of the different step sizes is compared to see if the results are linear. For the chosen step sizes all responses show linear behavior. From the four step responses of different step sizes at one operating point, the mean normalized response is calculated. Five mean normalized step responses are obtained for the five operating points. They are displayed in Figure 2.8. From Figure 2.8(b) it can be concluded that the system is time dependent, hence a different LTI model is derived for every operating point.
CHAPTER 2. MODELLING FOR AIRBAG CONTROL
-time [ s ] ( a ) Step responses at different operating points
0
0.02
0.04
0.06
time [ s ] (b) step responses displayed i n fig. ( a ) shifted in t o t = 0
Figure 2.8: Mean filtered normalized step responses. (rj [ms]) It turns out that the following choice yields the models with the least error between LTI output and crash model output. rj ~ t ~ 0 . 0 9 0 [ sfor ] rj=40,50,60 and 70 ms rj 2 t 5 0.098 [s] for rj = 80 ms
(2.26)
The singular values of the Hankel matrix containing the step response data, are displayed in Figure 2.9 and Table B.2. Again, the matrices A, B, C and D are translated into the standard second order transfer function of Equation 2.25. For the different step times, every time, two singular values are significantly larger, which means the order of the models to describe the dynamics, like for are of order 2. The resulting data of the transfer functions of the different step times is displayed in Table 2.2. For rj = 50 ms, the step response data of the realized state-space LTI models is displayed together with the crash model step responses in Figure 2.9(b). For completeness, the step responses of the other realized LTI models are shown in Figure B.2. Again, the time dependency of the model can be observed. Bode diagrams of the models are displayed in Figure 2.10. When comparing the LTI models of the different operating points like in the previous subsection, time dependency seems small again because their Bode diagrams are almost the same. Only for operating point rj = 80 ms, the Bode diagram differs from the rest in the low frequency part. Because there is no indication that the dynamic behavior is different at the end of the crash, it is assumed that the modelling process causes the difference. Therefore the model of rj = 80 ms is considered an outlier that can is not considered as important for controller design as the LTI models of the other operting points.
2.3. APPROACH APPLIED TO THE CRASH MODEL
- - LTI model resp.
-100 0.05
0.06
0.07
0.09
0.08
time [ s ]
Figure 2.9: Singular values and the step response of a approximate state-space realization together with the crash model step response
.rj = 40
w, [ H z ]
t
ms 1.7310-10' 2.7383-10-I
r j = 50 m
s 2.3235.10' 2.7793.10-I
r j = 60
ms 3.0224.10~ 2.8133.10-I
r j = 70
ms 3.6507.10' 3.4414-lo-'
r j = 80
ms 3.4321.101 1.9448.10-'
Table 2.2: Model data of the transfer functions from A, t o xh
Figure 2.10: Bode diagrams of the transfer functions from A, t o xh
CHAPTER 2. MODELLING FOR AIRBAG CONTROL
2.4
Discussion
The models derived using the approximate realization method, are able to give a good prediction of the step response of the crash model in the operating points. The error between the first part of the step response of the LTI models and the step response of the crash model is relatively small, for both +in as well as for he conclusion is that the crash model can be linearized around the operating point if the used steps are relatively small compared to the original input. Furthermore the time domain must be long enough to derive good models and small enough to prevent time dependency to influence the models to much. Time dependency of the crash is caused by changes in the direction of motion of vehicle and dummy and the end the fact that the acceleration of the head is zero at t = 100 ms. In closed loop, the reference trajectory will affect the operating points, therefore closed loop identification should be done to see if the resulting models correspond with the open loop derived models. The derived LTI models are all of second order and have almost the same eigenfrequency, damping and zero. Even when we compare the models of input +in with those of input A, we see that the only major difference is the gain. The time dependency seems to be larger in time than in frequency domain since the step responses vary whereas the Bode diagrams are more or less the same. When the LTI models are examined, the model of +in for rj = 40 ms is different from the other models and is considered as not representative for the actual dynamics. Because this model for input +in is at the beginning of the crash, a lot of reasons can cause the modelling to fail. It should be investigated if the models can be left out or if the dynamics are really different at the beginning and the end of the crash. For controller design in the next chapter, models of rj = 50 ms, rj = 60 ms, 7j = 70 ms and rj = 80 ms are considered to be more important. Therefore, the model of +in for rj = 40 ms is not taken into account during controller design.
Chapter 3
Control er Desi The derived LTI-models of the previous chapter are used to design a controller for input q5in and a controller for A, to control xh. During simulations, one input is manipulated and the other one is passive. For the passive input, the original time signal is used. This way the control scheme comes down to a single input single output (SISO) controller setup, shown in Figure 3.1. Input u(t)is either 4in or A, and output y ( t ) represents xh.
(a) for control design with LTI models
(b) for controller validation with crash model (behind the controller a saturation is added: u 2 0)
Figure 3.1: Closed loop with P ( s ) , the LTI models and M , the crash model. The limitation of this approach is that the pressure in the airbag can either be increased or decreased actively. As a result, in general the head acceleration can only be increased or decreased respectively.
with an increase defined by (T) and decrease by (l), the force between head and airbag Fh-b and the pressure inside the airbag pb. Because xh is defined as acceleration, an increase in restraint force results in an increase in deceleration and therefore, A x h ( L ) . However, because this is a straight-forward approach, a single input is used before other strategies are investigated that enable the use of both actuators. Furthermore, because the derived LTI models are very similar for every operating point, it is investigated if one controller can be designed that can be used during the total crash.
CHAPTER 3. CONTROLLER DESIGN From the original US-NCAP simulation, the head is in contact with the airbag from t = 35 ms. Things that need to be kept in mind are: 1. For t < 35 ms the original signals, q5in,o/A,,o, are used for both inputs. These inputs are used to get the airbag in a good state to receive the dummy's head. Once the airbag and head have stable contact, the controller takes over. 2. For t 2 35 ms one input is controlled. The reference trajectory is designed in such a way that it starts at t = 35 ms with the same acceleration as the original crash, xh.o. Although the contact area of the airbag and the pressure at t = 35 ms is still small and not suitable for identification, control is started to get as much time as possible to decelerate the head. 3. During all crashes, the dummy is wearing a seat belt that is not manipulated. In Section 3.1, the setpoint is determined. In Section 3.2, general control design is discussed. In Sections 3.3.1 and 3.3.2 this method is applied to design controllers for the two inputs of the airbag, q5in and A, respectively. Section 3.4 discusses the differences between q$, and A, manipulation. Finally, Section 3.5 shows closed loop identification. In Section 3.6, the results are discussed.
3.1
Reference Signal
A reference signal is needed for the acceleration of the dummy's head that is controlled to reduce the sustained injury during a crash. The method of [4] is explained roughly. From the original crash it can be concluded that the possible time period to control the head acceleration is 35 5 t 5 100 ms. The injury criterion HIC should be as low as possible to reduce the risk of injury. The maximum deceleration should be reduced as much as possible but it is obvious that the same energy needs to be absorbed as during the original crash. It is a plausible assumption that a constant reference signal with an as-low-as possible acceleration is appropriate. As a result, HIC can be reduced significantly compared to the original crash. To smoothen the transition region at the beginning and the end of the reference for the controllable part of the crash, a 3-th order profile is used because a controller has difficulty with discontinuous signals. There are also constraints that may not be violated. The head must stay within the fixed space between seat and steering wheel, lo in Figure 3.2. This means the head of the dummy is not allowed to move through the seat or to touch the steering wheel. At the end of the crash the head must have the same velocity as the vehicle. For a significant reduction of the maximum deceleration, the available space, lo should be used as much as possible. This is visualized in Figure 3.2 and it can be concluded that in this case the total distance is used. The position and velocity of the head are all considered relative t o the vehicle and the occupant is considered as one solid object of a certain mass. The
3.1. REFERENCE SIGNAL
r
occupant
occupant
i
occupant
I
Figure 3.2: Schematic representation of the position of the head relative t o the vehicle at dzflerent times during the crash criteria of the reference signal are:
with xh, the position of the head, xueh,the position of the vehicle/steering wheel, te = 100 m s is the end of the crash and lo = 0.44 m, the initial distance between head and steering wheel. It is assumed that the vehicle position, velocity and acceleration are known a priori. After some iterations to satisfy all criteria of Equation 3.1, the reference signal is defined as follows. The features of the 3th order signal are displayed in Figure 3.3. The constant value of the reference is 286 m/s2between and t = 40 m s and t = 94 m s . If the reference is followed perfectly, HIC is reduced from 570 to 166. This is a reduction of over 70%. In Figure 3.3, a visualization of the position, velocity and the acceleration of the dummy as a result of the reference signal is given. It can be seen that indeed all of the constraints of Equation 3.1 are satisfied.
time [ s ]
time [ s ]
(d
(4 Figure 3.3: setpoint
CHAPTER 3. CONTROLLER DESIGN
3.2
Controller Setup and Design Criteria
The goal is to design a stabilizing controller that forces the head acceleration to follow the reference trajectory, satisfying a certain set of stability-margin and performance specifications. The basic control setup for airbag control is displayed in Figure 3.1. Let u(t) be the input that is controlled (either & or A,) and P(s),one of the LTI models. The error is defined as e(t) = r(t) - xh(t) and the reference signal is r(t). The controller is denoted by C(s). To analyse stability and performance of a closed loop system, various transfers are used. The open-loop is defined as L(s) = C(s)P(s), the sensitivity is S(s)= l+C(:)P(r). The stability criteria are:
[-I.
e
The gain margin (GM) is 2
e
The phase margin ( P M ) is 45
e
The sensitivity is not allowed to cross the 6 dB boundary.
[O].
The stability margins GM and PM, which are visualized in the Nyquist diagrams, are relatively large because the LTI models do not offer any insight in the nonlinear dynamics of the crash model. To get a more robust controller the margins are defined relatively large. The 6 dB margin in the sensitivity is also an extra safety margin. An important performance criterion is error reduction. Steady-state error reduction can be determined from the sensitivity. If the maximum error is defined as 6% of the reference in the time domain, in frequency domain this means the magnitude of the sensitivity jS(s)! must be below -24.5 dB for the frequency contents of the reference signal. In Figure 3.4 the spectral density of the reference signal is displayed. It can
Frequency [ H z ] Figure 3.4: Power spectrum of the reference signal derived in Section 3.1 be concluded that the dominant frequencies in the reference signal are below 30 H z . Therefore, !S(s)!should be maximum -24.5 dB for frequencies below 30 H z .
3.3. CONTROLLER DESIGN AND IMPLEMENlXTION
The bandwidth, defined as the 0 dB crossing of the open-loop OL, needs to be determined. The choice of the bandwidth is always a trade-off between performance on one side and amount of effort on the other. Several factors such as computational noise and undermodelling lead to the desirability of decreasing the open-loop gain at high frequencies, thus putting a limit on the on the bandwidth of the closed loop. Although a Madymo simulation produces mostly high-frequent noise, also in the low-frequent region noise is present. For this reason the bandwidth is taken as low as possible. An indication for the minimum bandwidth can be found by looking at the steady state error criterion. From this design criterion, the amplitude of S(s) must be below -24.5 dB for frequencies below 30 H z . Together with the phase margin of 45', this can be used to obtain an approximation of the desired bandwidth. If the OL phase margin is 45', the
-30 1oO
I
I
1o4
102
Frequency [Hz] Figure 3.5: Sensitivity phase is -(I80 - PM) = -135'. This means the slope of the amplitude of S(s) is f1.5, In Figure 3.5, a sensitivity if it is assumed that C(s)P(s) >> 1 and S(s) zz m. plot is shown. Starting at -24.5 dB for f = 30 z with a constant slope of f1.5, IS(s)1 crosses 0 dB at f = 197 H z . At this point IC(s)P(s)1 = 1and thus S ( S C)( s@ A )P(s) Since this is only a rough approximation of the desired bandwidth, the frequency at which the magnitude of the sensitivity crosses 0 dB is still considered equal to the open-loop bandwidth. As a result we have an indication that the desired bandwidth is approximately 200 H z .
ii'
'
3.3
Controller design and implement ation
The Bode diagrams of the LTI models show a phase lead around 200 H z that can be used instead of a derivative term in the controller. A bandwidth of 200 H z is chosen. If only one constant controller is used, the bandwidth is not constant, but varies around 200 H z because of the different transfers of the LTI models. It must be analyzed if 200 H z should preferably be the mean, maximum or minimum bandwidth of all LTI models . Furthermore, it needs to be evaluated whether a controller exist that satisfies the design criteria for every LTI model of the various operating points.
CHAPTER 3. CONTROLLER DESIGN
3.3.1
$in
controller
A controller is designed using the LTI models of Section 2.3.3. The stability and performance criteria must be met for all LTI-models of the operating points. As already mentioned, the LTI model of 7j. = 40 m s is not taken into account. Because the Bode diagrams of the other LTI models for input bin do not significantly differ, the aim is to use one controller that satisfies the criteria. Due to the slightly different transfers of the operating points, the bandwidth cannot be defined by a single cross-over frequency. Therefore, the mean bandwidth for ali transfers is chosen to be 200 H z . The controller is build up out of an integral action with a low-pass filter. Finally the gain is adjusted in order to get a bandwidth of 200 H z (Equation 3.2). The integral action has a zero at 50 H z t o increase error reduction in the low-frequent region below 30 H z . The Low-pass filter is of order 2 with a cut-off frequency of 1000 H z to reduce the influence of high-frequent noise. The transfer function of the controller is:
Y
gain
integrator
'
-+
low-pass f ilter
with fi = 50 H z the zero of the integrator, flpf = 1000 H z the double pole of the lowpass filter, ,8 = 0.7 [-] and gain k = 1.27 . l o p 2 [-]. The Bode diagram of the controller is displayed in figure 3.6 The Bode diagram of the open-loop is displayed in Figure 3.6(b). The bandwidth varies between 138 H z and 270 H z for the different LTI models. In Figure 3.6(c) the sensitivity is displayed. The boundary of 6 d B is not crosses (the solid horizontal line in Figure 3.6(c)) and the magnitude stays below -24.5 dB for frequencies below 30 H z (the dashed horizontal line in Figure 3.6(c)). The stability of the designed control system can be analyzed in the Nyquist diagram in Figure 3.6(d), showing that the closed-loop system is stable and the desired GM and PM are achieved. Evaluation using the LTI models The controller is tested in the time-domain using Simulink with the linear transfer functions of the identification. The blok-scheme of the Simulink implementation is the same as in Figure 3.l(a) except that a saturation like in Figure 3.l(b) is used. Because of saturation on the inputs (+in 0) integrator windup will occur. A reset on the integrator when the error crosses zero, can prevent the windup.
>
Simulations are done for all four LTI-models (rj= [50,60,70,80] ms). Because they all show the same behavior, only one result is shown in Figure 3.7 (for rj = 60 ms). Note that this linearized model is obtained around the original input of +in,o and normally $in can vary between 0 and +m. Therefore, during the LTI model simulation the original input is subtracted from the controller output because it is already discounted in xh,0. For +in however this is not really relevant because do FZ 0 for t 2 35 ms. The results of the simulation are good. Between 35 and 40 ms, the amplitude of the tracking error stays within the specifications of the design criterion. Since input +in is controlled and A, is passive, only a negative tracking error can be compensated. This causes the response of xh in Figure 3.7. When the original response
3.3. CONTROLLER DESIGnT AND IMPLEIMENTATION
(b) Open-loop
(a) Controller
03
[Hz1
(c) Sensitivity
(d) Nyquist-diagram
Figure 3.6: Controller tuning is above the reference trajectory, the inflator can be used to move the head acceleration from the original acceleration towards the reference. However, between 40 ms and 80 m s a negative mass flow is needed to reduce the head acceleration but this input is passive. The inflator is active in the region from 35 till 40 ms and between 80 and 100 ms. After 100 ms, the deceleration of the head is not equal to the reference but the simulink simulation is stopped. The controller shows a satisfactory behavior in simulations with the LTI models. Evaluation using the crash model Now the controller is tested with the crash model. The closed loop system with crash model for the Madymo simulation is displayed in Figure 3.l(b). At t = 35 ms a vertical line is drawn to indicate Figure 3.8 shows the results for the time that the controller is activated. The results are almost equal to the simulation using an LTI model. That is a good result when it is taken into consideration that the complex nonlinear crash model can be controlled by a single PI controller yielding a
CHAPTER 3. CONTROLLER DESIGN
0
0.02 0.04 0.06 0.08 0.1
0.12
time [ s ]
time [ s ]
m3 2, x
1
'
I
300 1
1
'
time [ s ]
time [ s ] Figure 3.7:
manipulation using a n LTI model. At t = 35 m s , the controller is activated (dashed vertical line i n the figure).
+in
time [ s ]
01 0
time [ s ]
I
0.02 0.04 0.06
0.08 0.1
0.12
1
-1 00 0
I
v,
0.02 0.04 0.06 0.08
time [ s ] Figure 3.8: Crash model simulation using
time [ s ] +in
control
0.1
0.12
3.3. CONTROLLER DESIGN AND IMPLEMEN7XTION
stable closed loop system with a tracking error that applies to the design criterion. A better performance between 45 and 80 ms is only possible if a negative mass flow is possible. In the next subsection, first the same controller design procedure is repeated for input A,. Subsequently, the results of the two inputs are evaluated and compared together in Section 3.4.
3.3.2
A, controller
A controller is designed following the same steps like in Section 3.3.1. The bandwidth is chosen 200 H z . The controller is build up out of an integral action and a low-pass filter. Finally the gain is adjusted in order to get a bandwidth of 200 H z . The controller is equal to Equation 3.2, with fi = 60 H z the zero of the integrator for low frequent error reduction, fipf = 1000 H z is the double pole of the low-pass filter to reduce the influence of high frequent noise, ,6 = 0.7 and gain k = 1.27 . [-I. The controller is displayed in figure 3.9(a).
[-I
The Bode-diagram of the open-loop is displayed in Figure 3.9(b). The bandwidth varies between 100 H z and 225 H z . The gain is tuned so the models for rj = 70 m s and rj = 80 m s have a bandwidth closest to 200 H z . The bandwidth for the other models is automatically decreased because only one controller is used. For the A, controller this means that a compromise is necessary. The enlargement of the stability margins ensures that modelling errors and nonlinear effect have less influence, but on the other hand the performance decreases a bit. In Figure 3.9(c) the sensitivity is displayed. The stability of the designed control system can be analyzed in the Nyquist diagram in Figure 3.9(d) and shows that the closed loop system is stable but GM and P M are not satisfied for all operating points. For the approximate realizations at 7j = 40 and 50 ms, the phase margin is smaller than desired (but still at least 36'). However, it turns out after some simulations that the peak in sensitivity has to be as small as possible for the models of rj = 70 and 80 ms. For these models, the peak value of the sensitivity is 2.8 dB. The overall maximum peak value of IS(s)I is 4.2 dB. The reduction in sensitivity peak results in a reduction of the phase margin. As already mentioned during the modelling of the transfer from A, to xh, the Bode diagram of rj = 80 m s is different from the rest for low frequencies. It is considered a modelling error and therefore, the fact that the model does not apply to the performance criterion is considered not to be important. Evaluation using t h e LTI models The controller is tested in the time-domain in Simulink with the linear transfer functions of the identification. A similar controller setup is used as displayed in Figure 3.l(a) but with a saturation u(t) 2 0. Because of saturation on the input and the problem with integrator windup, the same integrator reset is used as in Section 3.3.1. The simulation are done for all five LTI-models ( ~ = j [40,50,60,70,80] ms). Because they all show similar results, only the result of the LTI model of operating point 7j = 60 ms is shown in Figure 3.10. Note that this linearized model is obtained around the original input
CHAPTER 3. CONTROLLER DESIGN
02
(a) Controller
(c) Sensitivity
[Hz1
(b) Open-loop
(d) Nyquist-diagram
Figure 3.9: Controller tuning. and normally A, can vary between 0 and +m (in theory). For the of A, = 1.73 . LTI model simulation the original input of A, = 1.73 - lop3 m2 is subtracted from the controller output and therefore A, in Figure 3.10 is A, 2 -1.73 . lop3 m2. This means that relative to the original acceleration, a "negative" vent size is possible which results in a increase of deceleration. Therefore, this case is not the opposite of what happened in Figure 3.7 which would mean that only for 45 5 t 5 80 ms A, can be active. Now only A, is controlled and &, is passive which means that only positive tracking errors can be reduced. But since the LTI models are derived around A, = 1.73 . the vent can be closed more than during the original simulation to reduce positive errors as well. Still a region remains in which the head acceleration cannot be influenced. Between 42 and 88 ms the A, is used to move the original response towards the reference trajectory with guaranteed stability and good performance. The vent has no influence before 42 ms or after 88 ms because then the error is negative.
Evaluation using the crash model Also for A,, the controller shows a satisfactory behavior in simulations with the LTI
3.4. EVALUATION OF THE TWO INPUTS
time [ s ]
time [ s ]
time [ s ]
time [ s ]
Figure 3.10: A, control LTI models. At t = 35 m s , the controller is activated (dashed vertical line i n the figure). models. Now the controller is tested with the crash model. The closed loop system with crash model for the Madymo simulation is displayed in Figure 3.l(b). Figure C.l shows the results for A, control. At t = 35 m s a vertical line is drawn t o indicate the time that the controller is activated. The tracking of the reference signal is very good. But because the reference cannot be followed during the first part of the crash with A, control, the total deceleration is smaller than the reference and the head will touch the steering wheel. The reference is changed in such a way that the head does not touch the steering wheel. It is multiplied by 1.05 t o make sure that, despite the loss in the beginning, the total deceleration is enough t o keep the head from touching the steering wheel. This result is shown in Figure 3.11. Now the tracking is good and the specifications of steady state error and not touching the steering wheel are met.
3.4 Evaluation of the two inputs In Figure 3.12, the kinematics of the dummy during the crash are displayed. In (a) for & and in (b) for A,. In (b), the airbag is removed from the picture for clarity and in (a) the airbag is displayed in wire-frame. Both (a) and (b) show the kinematics at the time that the head of the dummy is closest to the steering wheel. For & that is t = 70 m s and for A, t = 95 m s . The difference between 4i, and A, control is very obvious. Using a controller for input & leads t o little improvement relative to the original crash. The kinematics show that
CHAPTER 3. CONTROLLER DESIGN
x Io
-~
time [ s ]
time [ s ]
I
time [ s ]
time [ s ]
Figure 3.11: Crash model simulation using A, control. Repeated simulation of Figure This time, the alternative reference trajectory prevents the dummy 1 from colliding with the steering wheel. New reference: (r,,, = l.05rold).
(a) dummy closest t o the steering wheel during #in control (70 ms)
(b) dummy hits the steering wheel during A, control (95 ms)
Figure 3.12: Kinematics at various times during the frontal US-NCAP crash with a controlled airbag the minimum space between head and steering wheel is large and could be used more efficiently to decelerate the dummy. The result of controlling A, is, as already mentioned that the head hits the steering wheel. But at that moment the head is already decelerated to a relatively low velocity. The impact is not too severe. The HIC value of the two inputs are 590 and 174 for respectively input 4,, and A, control. Remember that the original and setpoint HIC are 569 to 166 respectively. Note that for controlling qhin the result is worse than the original crash. With this new reference for A, control, the results in both figures correspond with the
3.5. IDENTIFICATION ALONG CLOSED LOOP TRAJECTORY results obtained using the LTI models. When the limitations due to the fact that only a positive input is possible are taken into account, it can be concluded that the derived models and the tuned controller apply to the requirements.
3.5
Identification along closed loop trajectory
The approximate state space realization method is used to derive LTI models at a number of closed loop operating points. Since the head acceleration has t o follow a reference signal that is completely different from the original operating points, a closed loop identification is done to verify if the models are valid for controller design. At various points in time, differently sized steps are added to a time history of one of the inputs, obtained from a closed loop simulation. The step responses are obtained during open loop simulations using one of these input. In Figure C.2(a) and C.2(b), the response with a step in the input bin or A, respectively, can be seen for the operating point t = 70 m s . Also the original closed loop response is displayed as comparison. It can be concluded that the operating point is much closer to the state that the airbag will be in during closed loop simulations. A step in A, causes the head deceleration to drop and therefore the reference signal must be constructed such that the head does not hit the steering wheel despite the lower deceleration. The resulting models for the two transfers at t = 70 m s are displayed in Figure C.2(c) and C.2(d) and Table 3.5.
bin
'
( T ~= 70 m s ) 2.396-lo1 w, [Hz] 5.154-10-I C -1.050-lo3 k z [ H z ] -3.008.10~
A,(T~= 70 m s ) 2.427.10' 4.652-10-I 6.067.10~ -7.894
Table 3.1: Model data of the transfer functions for both transfers derived with closed loop inputs
The four Bode diagrams of the open-loop and closed-loop identification for both inputs are displayed in Figure 3.13 The difference in gain and sign between $in and A, can be seen. The phase of starts at -180°, whereas A, starts at 0' for low frequencies. A relevant difference between the open- and closed-loop identification however, is not present. By looking at the step response, the model parameters and the Bode diagrams, it can be concluded that the closed- and open-loop system identification yield approximately the same results. Final conclusion, the models of the open-loop identification are useful for controller design although the eventual trajectory is significantly different.
CHAPTER 3. CONTROLLER DESIGN
Figure 3.13: Bode diagrams of the transfer functions from +in t o xh and A, t o xh of both closed-loop (el) and open-loop (01) identification
3.6
Discussion
From the simulation results with the LTI models as well as with the crash model, it can be concluded that both inputs of the airbag can be used to control the head acceleration but a significant reduction of injury risk is achieved manipulating input A,. The results show that the approximate state-space realization modelling approach and the loop-shaping controller design are capable of designing a stable controller, which reaches the predefined criteria in the controllable region for head acceleration control during a crash using an airbag. When the error is positive or negative for +in or A, control respectively, the controller has no influence on the head acceleration. For +in control this effect is much worse and the overshoot is not an improvement with respect to the original situation. For A, control, the effect of a positively constrained input is less than for din and results in a good tracking of the reference between 45 and 90 ms. Because the acceleration is above the setpoint between 35 and 45 ms, the head hits the steering wheel after 90 m s but at relatively low forward speed. As a result of using only one of the two inputs for control, the constrained behavior of the airbag inputs causes the head acceleration to be uncontrollable for a period of time. This phenomenon is very clear when only +in is controlled (See Figure 3.8). At a certain point in time the pressure in the airbag is too high but it cannot be reduced because the vent is passive. The controller has to "wait" until the pressure is reduces enough to pick up the trajectory again. For +in control the HIC value is increased and for A, control the setpoint needs to be adjusted to prevent the head from hitting the steering wheel. In the next chapter it is discussed if a combination of both din and A, control can be used as a possible solution for these problems and also enlarge the adaptability of the airbag to differences in crashes and occupants. The performance for varying circumstances is not examined. An evaluation of possible approaches t o use, is done to realize a flexible system capable of adjusting in case of varying circumstances.
Chapter 4
In the previous chapter, only one variable was used to control xh. From the evaluation of the results, a combination of both inputs is supposed to yield a better performance and more flexibility of the airbag to adjust to different crash and occupant characteristics. In can be used to reduce negative tracking error and A, positive tracking general, input error. A combination of inputs offers the possibility to reduce all tracking error.
&
In this chapter, several approaches are presented to achieve this combination. The new approach is implemented in block C of Figure 1.5. Those are compared for the different approaches to decide which one is the best of the proposed strategies. The most simple solution to come to a combination of two controlled inputs is designing a controller for both inputs separately using loop shaping. First, it needs to be investigated if the two inputs do not influence each others result in to xh the head acceleration i.e. if the output of the LTI models of the transfer from and from A, to xh can be added to obtain the total head acceleration. The LTI closed loop scheme compared to the crash model setup is displayed in Figure 4.1. With M the crash model and C l ( s ) and C 2 ( s ) the controllers for &, and A, respectively.
&
If the crash model can be considered consisting of two parallel systems with no interaction, the representation of the crash model in Figure 4 . l ( b ) is valid ( P l ( s ) and P2(s) represent the LTI models for both transfers). Normally the output of one or more parallel
(a) closed loop crash model
(b)closed loop LTI model
Figure 4.1: Combined din and A, control. Controllers C l ( s ) and C 2 ( s ) are the contents of block C of Figure 1.5. The input of block C , e ( t ) and the outputs, ul(t) and u2 ( t ) can be recognized.
C H A P T E R 4. COMBINED MANIPULATION OF INPUTS linear independent systems can be added to get the total transfer. This rule cannot be applied to the LTI models of the transfer from &,to xh and A, to xh straight away, since it is not clear what happens inside the crash model M between the two inputs ul and u2 and output y in Figure 4.l(a). There might be a multiplication or other relation between the two inputs: $(t) = u1u2P ( s ) . Until now, only one input has been controlled and the passive input remained at the same value as during the identification so no nonlinear effects were present. Now both inputs are used and the influence on each others response is no longer trivial. To verify if the value of one input does not influence the output of the system resulting from the other input i.e., if the sum of the output of the two LTI models PI and P2 in Figure 4.l(b) is equal to the total head acceleration, several simulations have been performed. From these simulations it can be concluded that the schematic in Figure 4.l(b) does apply for the total airbag system. A distinction is made between four cases: a , b, c and d. Every case is divided into three subcases: 1 , 2 and 3 for input din, input A, and a combination of both inputs, respectively. To make plausible that the two transfers can be considered independent and linear, the following relation must hold:
The influence of steps at different times during the crash is investigated by comparing cases a to d. For a and d both step times are the same for #in and A,, 40 m s and 50 m s for a and d respectively. For b, r(q&,)= 40 m s and r(A,) = 50 ms. For c, T(&) = 50 m s and r(A,) = 40 ms. For case b, the results are displayed in this section, the rest can be seen in Appendix D. The following 4 x 3 simulations are done: b.
1. 2. 3.
Adin = 0.05kgls, Adin = 0.05kg/s,
T = 4Oms
T = 40ms,
AA, = 0.1575 , AA, = 0.1575 ,
T
= 50ms
T
= 50ms
*
output: aXh,l(t) output: AX^,^(^) output: A~h,3(t)
The results of case b is displayed in Figure 4.2. It can be concluded that Equation 4.1 represents the correct behavior of the dual input one output system. The respons Axh,3(t) is (Axh,1(t)+ Axh,2(t)) because both lines lie close to each other, especially compared t o the peaks of the two separate inputs. The simulation results show that Equation 4.1 holds and thus that Pl(s) and P2(s) can be considered independent. As a result, controllers can be tuned for both inputs separately.
4.1
Straight forward combination of both inputs
Since already two controllers have been tuned in Sections 3.3.1 and 3.3.2 for the two inputs of the airbag &, and A, respectively, a straightforward approach is to use a combination of those controllers. The control scheme is displayed in Figure 4.l(a).
4.1. STRAIGHT FORWARD COMBINATION OF B 0 T H INPUTS
time [s] Figure 4.2: Relative acceleration: Axh = xh - x ~ , ~ A method to evaluate stability for the dual-input single-output scheme of Figure 4.l(b), is the P Q method presented in [9]. This method uses the total open-loop transfer from e to y, OLsiso:
OLsiso = Cl (s)Pl(s)
+ c2(s)P2( s )
(4.2)
Although the assumption, used for the PQmethod, that one of the two transfers has a significantly higher gain in the high frequent region and vice versa, does not really apply to the two actuators of the airbag, the general P Q method is used. Still it is assumed that P2(s)has a larger magnitude at low frequencies and PI(s) at high frequencies. The stability and performance of the parallel system depends on the poles and zeros of OLsiso. Also closely related to the stability is the relative proportion of the output of PI and P2. If the two outputs are nearly the same, the performance is influenced negatively if the relative phase of the the outputs is approximately 180°. In this case the two outputs create an opposite effect and the total output will be disturbed. In our case, the controllers Cl and Cz are designed in Section 3.3, the closed loop stability can directly be analyzed. The result of dividing Equation 4.2 by Cl(s)Pl(s) is called
Figure 4.3: straightforward combination of
and A, control Madymo
CHAPTER 4. COMBINED MANIPULATION OF INPUTS P Q ( s ) and yields: P Q ( S )=
Cz(s)Pz(s) Cl ( s )PI ( s )
At the frequency that the magnitude of P Q ( s ) is 0 dB, the output of PI and P z ( s ) is the same. The phase of PQ (s) is the relative phase between Cl ( s )PI ( s ) and C2( s )P2( s ). At the o dB frequency of P Q ( s ) , the phase margin (180' - L ( C I P l )- L ( C 2 P 2 ) )must be at least 60' in order to get a stable system without output interference. Using straightforward geometry it is explained in [9] that OL,,,, is greater than one if the phase margin of Pl and P 2 ( s ) is larger than 60". The Bode plot of P Q ( s ) for the LTI models of +in and A, at T = 70 m s , is shown in Figure 4.3. According to the PQ-method the implementation of the two controllers should be possible without causing problems with respect to stability and performance. The control approach is tested during a crash simulation. The results of the implementation of the two controllers is displayed in Figure 4.4. It is a good result The dashed
time [ s ]
time [ s ]
time [ s ]
time [ s ]
Figure 4.4: Straightforward combination of simulation
and A, control. Results of a Madymo
horizontal lines show the design criterion with respect to the static error. The error applies to the standard between 43 m s till 86 m s . Before 43 m s the error is due to overshoot. The performance is bad towards the end of the crash but there is no obvious reason why this problem occurs when we look at the sensitivity of the open loop, using the PQmethod criteria. Maybe the dynamics change significantly at the end of the crash due to low speeds of the head and low pressure inside the airbag. Resulting in the fact that the LTI models are not sufficient towards the end of the crash. But with only one controller,
4.2. ERROR BASED SWITCHING
in the previous chapter, this problem did not occur. It is tried to model the dynamics at the end of the crash using the approximate realization method of Chapter 2. But the short available step response at t = 85 and t = 90 ms relative to the time constant of the dynamics, do not allow trustworthy models to be derived. More attention should be given to investigating the phenomenon occurring towards the end of the crash, in future research. Modelling the dynamics of the head and airbag interaction towards the end of the crash is an option but also an analysis concerning the computational limitations of the finite element and multi body simulation when contact forces between head and bag and the pressure inside the bag are small, could be done.
A possible explanation is that after approximately 86 ms the error is starting to stabilize around zero and the vent size starts to go to zero. This causes switching between the two inputs due to the high frequent noise. The interaction between the controllers might be the cause for the system to become instable (see Figure 4.4). Apparently, 4in met A, have opposite phase and a gain greater than one. They amplify each other causing an instable oscillation of h500 H z . Possible solutions to the problem are discussed in the following Sections 4.2 and 4.3.
4.2
Error based switching
In this case, the error e(t) is used as a switching signal to switch between either the control of @inor A,. If the error is positive, the vent is used. If the error is negative, the inflator is used. Due to the dynamics of the controllers, it is still possible that both inputs interfere. Therefore, a dead zone of 1g, when no controller is used, is introduced to separate the two inputs. This approach is displayed in Figure 4.5 and the switch S can be summarized as, el=e el=O el=O
e2=O e2=O e2=e
if if if
-1s
e 5-1 g e 5 0 g e >O $7
The dead zone is for @inand not equally distributed around zero, because from the SISO simulations in the previous chapter, it is clear that A, is the most important input. Only if the error is starting to decrease too much, input q5in is used.
Figure 4.5: Error based switch Results of a simulation using this method is shown in Figure 4.6. The tracking performance is good and stays good during the whole simulation. The problem of the previous section is solved. The m a c h u m error less than allowed and HIC = 169. A disadvantage
C H A P T E R 4. COMBINED MANIPULATION OF INPUTS
time [ s ]
time [ s ]
time [ s ]
time [ s ]
Figure 4.6: Madymo simulation results using error based switching of this implementation is that the switch in error causes the two inputs to work at the same time which is very inefficient. This is due to the fact that only positive or only negative errors are passed to the controller because that was the whole idea of this approach. Therefore, the integrator winds up. It is clear to see that around 60 ms the error is approximately zero and the value of A, remains constant because no negative error is passed to reduce the vent size. The airbag leaks to much air so the inflator compensates for the large hole in the airbag. A reset in A, at 60 ms is not an attractive solution. In that case, a totally closed vent results in a sudden enormous pressure rise which causes the head to leave the reference. Although the head acceleration follows the reference quit well, maybe it is possible to find a more useful solution with respect to the value of the inputs. Another criterion is added. The mass flow and vent size, which both result in an opposite head acceleration, should not be active at the same time. This way unnecessary actuator activity can be omitted.
4.3
Controller-output based switching
The sign of the tracking error is not the best criterion to switch between the two inputs. If the error is negative that does not necessarily mean that &, is positive and should be used. The vent should first be closed before the $in is used (that was the problem in section 4.2). Another possibility is to let the value of the inputs, 4in and A, decide what input should be used. In Figure 4.7, the control scheme is displayed. The switch S gets information of the outputs of controller ulc and uz, and the error e.
4.3. CONTROLLER-OUTPUT BASED SWITCHING This leads to the following strategy implemented in switch S: When the controller is activated at t = 35 ms, the two inputs ul and ua are set to zero. During control initializing, the first of controller outputs ul, and u2, that gets larger than zero, is used and passed on to ul, or uz, respectively. The other input remains zero. The active input is used until it becomes zero. Than the first input to turn nonzero is passed through to the Madymo simulation by S.
Figure 4.7: Controller-output based switch In addition, the input din is only used if the error is smaller than -1 g. This is done to separate the two inputs so they are not going to interfere like in figure 4.4. This approach ensures that the correct input is used because otherwise the error increases, the used input decreases and the other input increases and at the appropriate time, the inputs are switched. A disadvantage of this approach is that at the time of switching between inputs, the new input does not necessarily start off at zero. The controllers are active the whole simulation and due to a difference in dynamics, their output does not have to be zero at the same time. On the other hand, the values of the inputs at switching deviate not much from zero because they have the opposite effect and comparable dynamics and controllers. This is also seen when looking at the previous control implementation in Figure 4.6.
A simulation with the crash model shows a good performance. See Figure 4.8. The HIC value for this approach is HIC = 170.5. The results are good. Despite the possibility of discontinuities in the input due to switching, the tracking error complies to the desired criterion. The steady state tracking error stays within the desired range. Now also the inputs stay limited and are almost not active at the same time. In the beginning, the controller activates the inflator quite strong to move the acceleration from the original acceleration towards the steep reference signal. At t = 40 ms, the reference changes to the constant deceleration level. The result is that the inflator is switched of and the vent opened very fast but not quick enough to prevent a little overshoot. For the remaining part of the crash, a very small amount of extra gas that is supplied around t = 90 ms. Towards the end of the crash the error increases due to the interaction of the two controllers but in general, the error stays within the desired region. In Figure 4.9, the kinematics at two different times during the crash are displayed. It can be seen that the dummy does not touch the interior of the car. Furthermore, in Figure 4.9(c) can be seen that the whole space between initial position of the head and the steering wheel, lo, is used for deceleration of the head. This applies to the setpoint desigr, criterion.
C H A P T E R 4. COMBINED MANIPULATION OF INPUTS
-600
-
-
ref.
I
-7nn
time [ s ]
time [ s ]
time [ s ]
time [ s ]
Figure 4.8: Results of Madymo simulation with controller-output based switching
(a) position of the dummy at t = 64 ms)
(b) closest to the steering wheel (78 ms)
(c) zoomed in on (b)
Figure 4.9: Kinematics of controller output based switching
4.4
Discussion
A combination of two controlled inputs results in a better tracking of the reference signal than one controlled input in Chapter 3. However, if the two controllers are used simultaneously, the closed loop system with the crash model becomes instable. Several causes for this behavior can be given and it is not clear if the interaction of the two parallel controllers is the main reason. More research is needed to find out what the real cause is. For now, it is assumed that interaction is the main cause and therefore a switch is introduced to separate the two inputs. The advantage is that the two controllers can still be designed using loop shaping, for every input separately. The size of the dead zone
4.4. DISCUSSION
is determined in a few iterations using different values. It is an optimum between a small dead zone that still results in interaction and a large dead zone causing the tracking error to increase. The reason why 1 g is the optimum for this crash and what its value should be for other crashes, needs to be determined.
CHAPTER 4. COMBINED MANIPULATION OF INPUTS
Conclusions and ecornmenda The purpose of an airbag is t o decrease the risk of serious injury during a crash. In the introduction, the development of a strategy to design controller(s) for the airbag inputs to force the head to follow a reference signal, is defined as the objective of this thesis.
Conclusions The approximate realization method is chosen to derive dynamic models of the relevant transfers from mass-flow to head acceleration and from vent-size to head acceleration, required for controller design. The method is straight forward, gives information about the relevant order of the dynamics, results in simple linear and time independent models. Most important, the simulated step response of the LTI model, accurately predicts the response of the crash model to a perturbation. Closed loop identification shows that, for different operating points, the transfers are not significantly different compared to open loop. Loop shaping is used to design controllers. For both inputs, it shows that a single PI controller with a low-pass filter is able to satisfy the design criteria for stability and performance. However, the fact that u(t)> 0 makes it impossible to track the complete reference signal if only one input is manipulated. Still, loop shaping proved to be a straight forward and good way to derive controllers for an airbag. Designing the reference is easy and straight forward but can cause problems if, for some reason, the reference cannot be followed for a certain time. A consequence can be that the occupant hits the steering wheel. Controlling both inputs instead of only one can improve tracking since then, the whole reference signal can be followed by the head acceleration. Of the proposed approaches for a combination of inputs, the controller output switch yields the best results. To keep the two controllers from interfering when the error is around zero, the inflator is only used if the error is smaller than a certain negative value, whereas the vent is controller if the error is larger than 0 g. This control approach is a good way to implement the controllers of both inputs simultaneously. To evaluate the performance of this approach, the HIC value of the original crash is compared to that of the new airbag control approach. The HIC value is reduced by 70%,
CONCLUSIONS AND RECOMMENDATIONS from 570 to 171. It can be concluded that feedback control is capable of forcing the head to follow a predefined reference.
Recommendations
The new approxh is applied to the 50th percentile dummy at the driver side during the US-NCAP frontal crash. To test whether the method is also applicable for different characteristics, the approach must be evaluated for different occupants, crashes and airbags. Practical problems need to be solved before this approach can be implemented for an arbitrary crash, e
If the models are different for various vehicle and crash characteristics, the controller is probably tuned on stability and performance might decrease. If it turns out that the controller needs to be adjusted for different characteristics, an algorithm is needed to make sure the correct controllers are used. It might be possible to collect models for a number of representative circumstances and compare them t o see whether or not this is necessary.
e
It can be difficult to determine the right reference. It must be determined before the crash, but depends on the kind of crash. Therefore, the first few milliseconds of a crash are crucial to determine the correct reference signal. A possible approach based on the initial velocity can be a good strategy to design the reference designed for the considered US-NCAP crash. The time to start control needs to be variable. However, maybe the traditional concept of inflating the airbag must be replaced by a new approach in which the airbag is inflated as fast as possible with closed vent. Control of the inputs of the airbag is started as soon as the head makes contact with the airbag.
The cause of the problems in Section 4.1 must be determined. During the last part of the crash the system becomes instable. Possible causes could be changing dynamics or numerical problems. In Chapter 4, a dead zone of e = 1 g is the optimum space to separate the two inputs for this crash. This value is determined iteratively. It should be investigated if and why this value is correct and whether it changes for other crash circumstances. Before this approach can be implemented in a real passenger car, a lot other problems will need t o be solved like detecting the kind of crash and occupant. As can be seen from the closed loop simulation with the crash model, the actuators must have a large bandwidth with relatively large amplitude.
[I] Bendjellal F., Walfisch G., Steyer C., The programmed Restaint System from accidentology, SAE 973333, Collission Safety Engineering, 1997.
-
A lesson
[2] Chan C. Y., A Treatise on Crash Sensing for Automotive Air Bag Systems, IEEE Transactions on Mechatronics, vol. 7, no. 2, June 2002. [3] Franklin G.F., Powell J.D., Amami-Naeni A., Feedback control of Dynamic systems, 3rd edition, Addison Wesley Publishing Company, 1994. [4] Hesseling R.J., Steinbuch M., Veldpaus F.E. and Klisch T., Control Design of a Safety Restraint System, Proceedings of the 4th Asian Control Conference, Singapore, 2002. [5] Ho B.L., Kalman R.E., EjSCective construction of linear state-variable models from in- put/output functions, in Proceedings of the 3rd Annual Allerton Conference on Circuit and System Theory (Monticello, Illinois, Oct. 1965) (M.E. Van Valkenburg, ed.), pp. 449-459, 1965. [6] Huston R. L., A Review of the EjSCectiveness of Seat Belt Systems: Design and Safety Considerations, International Journal of Crashworthiness, vol. 6, no. 2, 2001. [7] Kramer F., Passive Sicherheit von Kraftfahrzeugen, Braunschweig, Germany: Vieweg, 1998. [8] Kress T.A., A Discussion of the Air Bag System and Review of Induced Injuries, SAE 960658, Tenessee, 1996. [9] Schroeck S.J., Messner W.C., McNab R.J., On Compensator Design for Linear Time-Invariant Dual-Input Single-Output systems, IEEE/ASR!tE Trans. Mechatron., vol. 6, NO.l, pp. 50-57, March 2001.
[lo] Schutter B. de, Minimal state space realization in linear system theory:
An overview, Journal of Computational and Applied Mathematics, Special Issue on Numerical Analysis in the 20th century - Vol I: Approximation Theory, vol 121 no 1-2, p.331354, Sept. 2000.
[II] Peeters Weem B.P., Airbag Manzpulation: Identi$cation and control, DCT-report 2003-19,Technische Universiteit Einhoven, Eindhoven, The Netherlands, Apr. 2003. [12] Rutten S.H.L.A., Literature Survey: State-of-the-Art of Passive Safety, Crash Safety Centre, TNO, Delft, The Netherlands, Dec. 2003.
BIBLIOGRAPHY [13] Silverman L.M., Realization of linear dynamical systems, IEEE Transactions on Automatic Control, vol. 16, no. 6, Dec. 1971. [14] Wang, J.T., Nefske, D.J., A new CAL3D Airbag inflation model, SAE 880654, Detroit, 1988. [15] Youla D.C., Tissi P., n-port synthesis via reactance extraction - Part I, IEEE Internat,ional Clnnvent,ion Record, vol. 14, pt. 7, pp. 183-205, 1966.
[18] Trafic Safety Facts 2002, NHTSA 2002, (http://www-fars.nhtsa.dot.gov).
[20] Madymo (http://www.madymo.com, Madymo manuals), TNO Automotive, Delft, The Netherlands. [21] Lecture notes 4K560, Physical Modelling for Systems and Control.
Appendix - A
elt syste
A.l
Airbag components
Frontal crash airbag system. The airbag module consists of an inflator (or gas generator) with an initiator, a textile bag ("cushion"), housing and, for driver bags, a cover for the steering wheel (see Figure A.l). The most common inflators use solid propellants, while hybrid inflators use a combination of compressed gas and a solid fuel. The cushion is made of nylon and folded in a specific way to make it unfold fast and safely. It has vent holes on the underside to assure the correct soft landing ("ride down") of the car occupant in the bag. The cushions are either sewn from purchased fabric or made directly from the yarn on the loom with a one-piece-weaving technology. The size of the cushion varies from 35 to 70 liters for the driver side airbag and from 60 to as much as 160 liters for the passenger side airbag.
housing
inflator
folded bag
(a) Driver side
(b) Passenger side
Figure A. 1: Driver Airbag (Autoliv Inc. [I 91)
APPENDIX A. AIRBAG AND BELT SYSTEM COMPONENTS
Figure A.2: Load limiter (Autoliv Inc. [I 91)
The housing is usually made of steel, but Autoliv has also introd3aced housings in strong, light-weight plastics. The cover over the driver airbag is made of plastics. It is forced opened by the pressure from the deploying bag. The cover has a split line t o make it open at a low pressure and "hinges" to keep its doors in place. The bag is fully inflated within 50 thousands of a second the eye - and deflated within two tenth of a second.
A.2
Seat belt systems components
A.2.1
load limiter
-
half the time of the blink of
Load limiters absorb the load in a crash in a very efficient way by keeping the belt force a t a controlled and pre-defined level. This is accomplished by a mechanism in the retractor that allows webbing to be pulled out slightly - and in a controlled way - if the load on an occupant's body becomes too high in a violent crash (see Figure A.2). The system is typically used in combination with an airbag which then absorbs the excessive energy. This is especially important for elderly, since studies have shown that a 60 year old person can only take half as much load on his rib cage as a twenty year old person. The latest generation load limiter is integrated into the retractor, where a specially designed bar holds the spindle with the webbing. As long as the force from the webbing exceeds a pre-set limit (usually 4kN), the end of the bar will turn, twisting the bar and thereby gradually reducing the load on the occupant's chest. Seat belt systems with load limiters typically also have pretensioners that reduce the forward motion of the occupant by tightening the belt in a crash. To further optimize the consistent retraining load from seat belts and airbags, Autoliv has started to introduce load limiters that work in two steps. In the initial onset of the crash, when the occupant is only restrained by the belt, the restraining force of the seat belt is held at a relatively high, constant level. As the occupant moves forward and into the airbag, the seat belt's load limiter switches to "the second gear" a lower restraining force that will prevent the risk of peak load that could occur if the restraining forces of the two safety systems were added t o each other. The 2-stage system therefore gives a high and relatively even load on the occupant's chest during the whole crash.
A.2. SEAT BELT SYSTEMS COMPONENTS
Figure A.3: Pretentioner (Autoliv I%c.[19])
A.2.2
Pretentioner
To make sure a seat belt restrains an occupant as early as possible in a crash, thereby reduce the load on the occupant in a violent crash, Autoliv has developed pretensioners. These tighten the belt during the very first fractions of a second in a crash. Pretensioners also reduce the risk of "submarining" (the car occupant slip under a loosely tightened seat belt). Pretensioners typically use the same sensor as the airbag. The two systems can then be tuned to maximize the protection for the occupant. Depending on the slack in the seat belt system, pretensioners can tighten the belt up to 15 cm (6 inches) by using one gram of a pyrotechnic propellant, either by pulling the seat belt buckle towards the floor or by operating the retractor. In Figure A.4, the subsequent stages of an ignited pretentioner are displayed. The generated gas turns the spool and tightens the seat belt.
A.2.3
Retractor
Retractors have two sensors that work independently on the locking mechanism. The vehicle sensor detects sudden deceleration of the vehicle, while the webbing sensor detects violent pull-outs of webbing from the retractor. A picture of a typical retractor is displayed in Figure A.4. Many future vehicles will have very advanced occupant weight sensing systems. In these vehicles, it will be possible not only to adjust the seat belt load t o the severity of the crash, but also individually to each occupant.
Figure A.4: Retractor (Autoliv Inc.[19])
57
APPENDIX A. AIRBAG AND BELT SYSTEM COMPONENTS This is an important advantage since smaller, lighter weight occupants, such as many women, are more susceptible to high belt loads than the average person, and these individuals do not need the same restraining force as a larger occupant. Autoliv's latest seat belt enhancement could therefore become especially important for female occupants.
Source: Autoliv Inc.
Appendix B
Additional figures for Chapter 2 For completeness, the singular values of the identification of the Hankel matrices used for the approximate realization method are presented in Tables B.l and B.2. al(7-j = 40 ms)
al(q = 50 ms)
al( r j = 60 ms)
a1 ( r j = 70 ms)
c1( r j = 80 ms)
1.0777.10~
7.5650.10~
5.5352.10~
4.2688~10~
1.8386-lo4 1.2580.10~ 4.1234.10~ 4.0243-lo2 3.2253-lo2
Table B.l: Singular values of T,,,l for
4i,
identi$cation
1 / g l( T j = 40 ms) o1(rj = 50 ms) I ci(7)= 60 n s ) I ci( ~ =j 70 ms) / ci (7-j = 80 ms) 1 I 2.9286.10~ 2.3562~10~ I 1.5536.104 I 6.6066.10~ I 3.1372.103
Table B.2: Singular values of T,,,I for A, identzfication The rest of the figures, showing the mean normalized step response of the crash model compared to the step response of the derived LTI models for the different operating points, are displayed in Figures B.l and B.2for input 4i, and A, respectively.
APPENDIX B. ADDITIONAL FIGURES FOR CHAPTER 2
11
Madymo resp. LTI model resp.
Madymo resp. LTI model resp.
time [ s ]
time [ s ]
(b)
-
- LTI model
- Madymo resp. - - LTI model resp.
resp.
-200
-1 200 0.06
0.07
0.08
0.08
0.09
0.09
0.1
time [ s ]
time [ s ]
200 - Madymo resp. - -
s
-200
i E x
-400
LTI model resp.
Y
time [ s ]
Figure B.l: Output of the derived LTI model compared t o the crash step response for transfer from din t o xh
rj
-
400
= 40 ms
rj
= 50 ms
400
-
-
300 -
N
300 -
w
- Madymo
- Madymo resp.
0.04
0.05
0.06
0.07
0.08
0.09
0.05
0.06
resp.
0.07
0.08
0.09
time [ s ]
time [ s ]
(4
(b)
- Madymo resp. - - LTI model resp.
time [ s ]
time [ s ]
(4
(4
time [ s ] (el
Figure B.2:Output of the derived LTI model compared t o the crash step response for transfer from A, t o xh
APPENDIX B. ADDITIONAL FIGURES FOR CHAPTER 2
Appendix C
itional figures for Cha
time [ s ]
time [ s ] 6
-
x 1o
-~
5
7
N
4
f -3
4
2 1 I
0 0
0.02
0.04
0.06
time [ s ]
0.08
0.1
0.12
-150
0
0.02
0.04
0.06
0.08
0.1
0.12
time [ s ]
Figure C.l: Crash model simulation using A, control. D u m m y hits the steering wheel at t = 90 m s . This is because the loss of energy dissipation during the first part of the crash.
APPENDIX C. ADDITIONAL FIGURES FOR CHAPTER 3
- - original response -500 -
original response step perturbated resp.
- step perturbated resp.
-600
°
0.04
0.06
0.08
0.1
0.12
-600
0
0.02
0.04
0.06
0.08
0.1
0.12
time [ s ]
(a) head acceleration with and without step perturbation 4in
(b) head acceleration with and without step perturbation A,
-S > 0
0.02
time [ s ]
Madymo stepresponse LTI model stepresponse
r
'
0
0
'
-1500 0.07
I 0.08
0.09
0.1
0.08
0.09
time [ s ]
time [ s ] (c) normalized head acceleration
LTI model stepresponse
-1 00 0.07
4i,
(d) normalized head acceleration A,
Figure C.2: Closed loop system identification
0.1
Appendix D
itional figures for Chapter 4 The following 3 x 3 simulations are done: a.
C.
A&, = 0.05kg/s,
T = 40ms,
1.
A& = 0.05kg/s,
T
2. 3.
d.
Adi, = 0.05kg/~,
1. 2. 3.
1. 2. 3.
A&, A&
= 0.05kg/~, =
-0.05kgls,
&bin = -0.05kg/s,
r = 4Oms
T
r
= 40ms
==+
,
T
= 40ms
,
T
= 40ms
==+ ==+
output: Axh,l(t) output: A ~ h , ~ ( t ) output: A ~ h , ~ ( t )
==+
output: A x h , ~ ( t )
, ,
AA, AA,
T = 40ms
+
= 50m.s = 50ms,
= 0.1575 = 0.1575
T =4Om~ 7
==+
output: A x h , ~ ( t ) output: axh,a(t) output: Axh,s(t)
AA, = 0.1575 AA, = 0.1575
= 40ms,
AA, = -0.1575 AA, = -0.1575
, T = 40ms ==+ output: Axh,z(t) , r = 40ms ==+ output: Axh,s(t)
From Figure D.l, it can be concluded that the assumption that the added outputs of Pl(s) and P2(s)represent the total head acceleration xh,is valid.
APPENDIX D. ADDITIONAL FIGURES FOR CHAPTER 4
time [s]
time [s]
(b) case c
(a) case a
time [s] (c) case d
Figure D.l: Relative acceleration: Axh = xh - xh,o
Appendix E
Symbols
frequency sample frequency time start of the crash end of the crash deceleration time period time horizon used for the minimal state-space realization gas mass in the airbag ambient air pressure pressure in the airbag mass-flow into the airbag
din
original mass-flow total mass-flow out of the airbag mass-flow through the vent of the airbag mass-flow through the porous airbag fabric dout = bent
+ &ores
Av
vent-size
Aw,o
original vent size (1.734945 -
M
Madymo crash model
P
LTI model
C
controller
APPENDIX E. SYMBOLS
output dynamic model reference
r (t>- Y ( t ) input gain margin phase margin size of step response data set gain difference factor gain undamped natural frequency relative damping zero system order time that perturbation is added unit step function scaling factor for the step function number of step size A number of step time
T
singular value number of singular value a xveh
position of the vehicle
xh
position of the head
~h
velocity of the head
x h , ~
original velocity
xh
acceleration of the dummy's head
xh,~
original acceleration
xh,s
acceleration as a result of step input
HIC
head injury criterion
lo
initial distance between head and steering wheel
ac
constant acceleration level of r ( t )
Fh-b
contact force between head and airbag
