Reduction of the chest deflection : a control approach: applied to a UNSCAP crash test with the BMW E46 driver

Reduction of the chest deflection : a control approach

Citation for published version (APA):

van der Zalm, G. M. (2003). Reduction of the chest deflection : a control approach: applied to a UNSCAP crash test with the BMW E46 driver. (DCT rapporten; Vol. 2003.009). Technische Universiteit Eindhoven.

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Supervision

ir. Rogier Hesseling

prof. dr. ir. M. Steinbuch

Reduction of the chest deflection:

A control approach

Applied to a USNCAP crash test with

the BMW E46 driver

G.M. van der Zalm

DCT

-

2003.09

Eindhoven University of Technology

Department sf Mechanical Engineering

Control Systems Technology

dr.

Sven Link

BMW AG

Advanced Safety Engineering

FIZ, EG-22

Contents

Abstract

1

Introduction 1.1 Today's approach 1.2 Proposed approach 1.3 Report structure 2 Modelling 2.1 Analytical model

2.2 110 estimation via stepwise perturbations 2.3 I/O estimation via stepwise inputs 2.4 Conclusions and discussion

3 Controlling the chest deflection

3.1 Introduction

3.2 Controller design with loop-shaping 3.3 Controller design with robust control

4 Discussion

4.1 Combining belt and airbag

4.2 Chest deflection versus chest acceleration

5 Conclusions and recommendations Bibliography

Appendix A: Extended Approximate Realization Appendix

B:

Small gain theorem

Abstract

In the last decades, extensive research has been performed in the field of safety restraint (SR) systems, i.e. systems that are activated when a crash is detected. Here, only one injury criterion is considered, namely the chest deflection of the driver. The chest deflection is defined as the displacement of the sternum bone with respect to the back of the dummy. This is done for only one crash type, the US- NCAP frontal crash test. Further, the airbag is neglected, so the SR consists of only the three point belt. The goal of this report is to minimize the maximum value of the chest deflection of the driver in a USNCAP frontal crash test with the BMW E46.

The minimization problem is translated into a tracking problem, by specifying a reference tra- jectory. This reference trajectory is derived with an analytical model, and is considered as the best possible solution. The problem can be stated as:

Design a controller which manipulates the belt force such that the chest dejlection of the driver in a

USNCAP frontal crash test with the BMW E46 tracks the reference trajectory.

First, a model of the transfer between belt force and chest deflection is made. Next, a controller is designed, using PID and robust controller design. The results can be used to design a smart SR system, but can also be used as an expedient for the design of the current SR systems. Then, a discussion about the influence of other injury criteria and using an airbag follows, and finally conclusions are drawn.

Chapter 1

Introduction'

In the last decades, extensive research has been performed in the field of safety restraint (SR) systems, i.e. systems that are activated when a crash is detected. A safety restraint system is designed with the use of crash tests, prescribed by law or by consumer organizations. The goal of these crash tests is to determine the quality of a SR system. Here, only one injury criterion is considered, namely the chest deflection of the driver. The chest deflection is defined as the displacement of the sternum bone with respect to the back of the dummy. This is done for only one crash type, the USNCAP frontal crash test. The airbag is neglected, so the SR consists of only the three point belt. Now, the goal can be defined as:

Minimize the maximum value of the chest dejection of the driver in a USNCAPfvontaL crash test with the BMW E46.

1.1

Today's approach

Todays actuators in a SR system are a compromise. Current actuators can only be activated once and they are adaptive only in the sense that the activation is chosen from a very limited set of possible activations. The choice of a specific activation is based on the most relevant crash characteristics directly after an impact is detected. After being activated, the actuators are not able to react or adjust according to newly available information.

The possible activations are chosen in the design phase of the SR system. Here, biomedical (e.g. shoulder-belt contact force), practical as well as technical restrictions are to be taken into account. The tendency in research in activation of SR systems is towards smart SR systems, which offer online adjustment possibilities, based on newly available information. A more appropriate restraining process of the occupant during the crash should then be possible.

Real crash tests are extremely expensive. New concepts for a SR system are therefore developed and tested using numerical simulation, for instance with the multibody and finite element package MADYMO, [l]. The car, dummy and crash set-up in this report are modelled with this package. The model consists of approximately 150 rigid bodies, and 150 (non)-linear springs and dampers, see figure 1.1.

Figure 1. 1: MADYMO model

1.2 Proposed approach

In this report, the maximum chest deflection of the driver in a USNCAP crash test with the BMW E46 is minimized, using the strategy as proposed in [2]. The minimization problem is translated into a traclung problem, by specifying a reference trajectory. This reference trajectory is derived with an analytical model, has to reflect a minimal maximal chest deflection, and is considered to be the best possible solution. Then, identification is done to obtain a simple model. Then, a controller is designed, to track the reference within predefined specifications. In the short term, the results can be used as an expedient for the design of the current SR systems. In the long term, the controller could be implemented as hardware. Since only the belt is considered, the only variable that can be controlled is the belt force. With this approach, the problem can be stated as:

Design a controller which manipulates the belt force such that the chest deflection of the driver in a

USNCAP frontal crash test with the BMW E46 tracks the reference trajectory.

The controller will be implemented in MADYMO using a MatlabISimulink coupling, with the control

chest deflection

9

setup of figure 1.2.

Figure 1.2: Simulink-MADYMO coupling

1.3

Report structure

MADYMO

reference + error

l n

,

>

First, a model of the system to be controlled is needed. This modelling will be done in chapter 2. Next, a controller can be designed. Two methods to obtain a controller are discussed in chapter 3. Then, a discussion follows in chapter 4, after which the conclusions are drawn.

Chapter

2

Modelling

The MADYMO model is not suitable for controller design, because of its complexity. Therefore, a less complex model is required. In this chapter, two ways to obtain a less complex model are discussed. To obtain a qualitative insight, an analytical approach is used. For a quantitative correct model, input- output estimation is used, based on step responses of the system.

2.1 Analytical model

2.1.1

Description force path in MADYMO model

Important for the modelling of the transfer from belt force to chest deflection, is the force path. The belt is pictured in figure 2.1. The force is applied to the belt at the downside of the B-pillar. From there,

Figure 2.1 : Beltpath in MADYMO

the belt passes over the D-ring on the topside of the B-pillar. Then it passes over the left shoulder, then over the sternum to the buckle-pretensioner on the right side of the seat. From there, it passes to the anchorfitting. It can be concluded that many force interactions take part in the transfer from applied force to chest deflection. Other forces are acting on the upper body, such as inertial forces and interaction between the dummy and the seat. These forces act on the upper torso, which roughly consists of three parts: upper torso, ribs, and sternum. The sternum is connected to the ribs, which in turn are connected to the upper torso, all represented by non-linear springs and dampers in the MADYMO model.

2.1.2 Modelling

Although the belt-dummy interaction is quite complex, to understand the behaviour a simple model can be made. The following assumptions have been made:

The chest deflection is caused by the belt force at the sternum.

The force on the sternum is the same as the force, applied to the belt on the downside of the B-pillar.

The interconnections between the torso, ribs and sternum can be modelled by linear springs and dampers.

e The interconnections experience no hysteresis and friction. The belt force is the only external force acting on the sternum. The various bodies can be considered as point masses. The problem can be considered one-dimensional.

o The upper-torso mass is much higher than the masses of ribs and sternum, so it can be consid-

ered as the fixed world for rib and sternum.

Then, the transfer from force to chest-deflection can be modelled by a mass-spring-damper system, pictured in figure 2.2. The sternum mass m, is connected to the rib mass m, by a linear spring with stiffness

k,

and a linear damper with damping coefficient

b,.

The rib mass is connected to the fixed world by a linear spring with stiffness

k,,

and a linear damper with damping coefficient b,. The

Figure 2.2: Two mass model

position of the sternum is described relative to the ribs, and the rib position is described relative to the fixed ground. Now, the equations of motion can be written:

The system is fourth order, since it is described by these two second order differential equations.

2.1.3 Evaluation

The values of the masses and the stiffness and damping coefficients, can be found in the MADYMO- data file, and are displayed in table 2.1. Non-linear stiffness and damping coefficients are linearized. The results of the model, according to equation 2.1 are compared with two simulations of the MADYMO-model: with the passive restraint system, and with the controller, as described in 121. This is shown in figure 2.3. From figure 2.3, it can be concluded that the sternum displacement

x,

according to this model corresponds very well to the result of the MADYMO model with the passive restraint system. In the second situation is the MADYMO result better predicted by the rib displacement

x,.

I

Parameter

/

Value

I

Unit

1

Table 2.1 : Model parameters br b~ m, m s 0.04 Time [s] 0.12, I - MADYMO 1 600 800 1.2 0.3 0.05 0.1 Time [s] Nslm Nslm kg kg

(a) Using force from passive RS (b) Using force from controller Hes- seling

Figure 2.3: Comparison of chest deflection in analytical model and MADYMO

2.1.4

Simplification

From figure 2.3, it can be seen that the rib and sternum displacement have a similar shape. This means that it might be possible to describe the transfer by a second order system. To investigate the dynamical behaviour further, the relative displacements between the bodies as calculated by MADYMO, are plotted in figure 2.4 (a). Furthermore, the forces which act between the torso and the ribs are given in figure 2.4 (b). The following observations can be made:

0 The rib displacement is the main cause (approximately 98 %) of the chest deflection. The dis-

placement of the sternum relative to the ribs is very low. Apparently, in the MADYMO compu- tations, the stiffness between sternum and ribs is very high. This means, that the model can be further simplified by considering the connection between sternum and ribs as rigid.

0 Since the relative velocities of the bodies are relatively low, the main part of the resultant force

is due to the spring force (approximately 80 %).

By considering the connection between sternum and ribs as rigid, the model is reduced to a single mass-spring-damper system, connected to the fixed world and excitated by the belt force, see figure 2.5. The parameters for this model are: m = m,

+

m, = 1.5 kg, I% = 5, = 1.5e5 Nlm, b = b, = 600

(a) Relative displacements (b) Forces between torso and ribs

Figure 2.4: Investigation in MADYMO

Figure 2.5: mass-spring-damper model

Nslm. Using the general differential equation form, the equation of motion becomes:

= 0.63, wo =

fi

= 316 radls and e = l / k = 6 . 7 1 0 ~ ~ m/N. Again, the model with ( = --

2&

is simulated and compared with the results of a simulation with the passive restraint system and the controller from [2]. The results are given in figure 2.6. The simplified model predicts the chest deflection qualitatively good. The prediction is quantitatively worse than the prediction with the two- mass model. By fitting the parameters, a good fit could be obtained for the two situations. However, the practical value of this approach is low, since the model has to be fitted each time for a different force situation. Nevertheless, this model gives a good insight in the dynamical behaviour of the considered transfer.

2.2 Input-output estimation via stepwise perturbations

Another way to obtain a model is fitting using input and output sequences. This method estimates a model, based on the impuls response of the considered system. Then, with approximate realization [4], a model can be estimated. Using this approach requires no physical background, which is an advantage if not enough information is available about the considered system, or if it is too complex to be modelled accurately.

Time [s] Time [s]

(a) Using force from passive RS (b) Using force from controller Hes- seling

Figure 2.6: Comparison of chest deflection in mass-spring-damper model and MADYMO

2.2.1 Approach

As said before, the method needs an impuls response. It is impossible to create an impulse in MADYMO, because this is a discrete program. Approximating an impuls in a discrete program re- quires a very small time step t,, and an amplitude equal to

$.

This will probably induce numerical problems in MADYMO. To overcome this problem, a step response is used, and the extended method is used, as described in [4], and in appendix A. By using a step response, low frequencies are weighed heavier, because of the low-pass frequency characteristic of a step response. This is an advantage, since high-frequent noise will have less influence.

Only the dynamical transfer from belt force to chest deflection is relevant. To avoid excitation of unwanted dynamics, the transfer from a small perturbation in the belt force to the corresponding response in chest deflection is considered as the transfer to be estimated. The aim is a transfer model which is valid along the passive force trajectory, so operation points are defined along this trajectory. Small perturbations are superposed on the passive force, and the step response is measured. To gain insight in linearity and time-invariancy of the dynamical behaviour, the steps are applied with different amplitudes and at different time instants. Then, the extended method of approximate realization can be used to obtain a model. With singular value decomposition, insight in the relevant model order can be obtained. In summary, the following steps are needed to obtain a model:

1. Obtain the belt force from the MADYMO-model with the passive restraint system.

Apply this force via the Matlab coupling, to check whether the coupling is implemented correctly. 2. Superpose stepwise perturbations on the belt force with varying amplitudes and at various time instants.

3. Check linearity and time-invariancy of the system, by investigating the step responses. 4. Reduce model order by investigating singular values.

Extended approximate realization [4]

5. Construct model.

2.2.2 Implementation

Obtain passive belt force

First, the passive belt force F o ( t ) should be obtained from the model with the passive restraint system. A simulation with the passive restraint system is done, from which the passive belt force is obtained. This is pictured in figure 2.7. Next, this force is applied via Simulink to a MADYMO-model with a

Figure 2.7: belt force in passive model

joint actuator. This gives the same results as with the original model, so the coupling is implemented correctly.

Superpose stepwise perturbations on passive belt force

To check wether the system is time-invariant and linear, the steps are superposed on different points in time, and with different amplitudes. A system is said to be linear, if:

~ ( t ) = L ( u ( t ) )

Then: a y ( t ) = L ( a u ( t ) )

and: L ( a u ( t )

+

b u ( t ) ) = a L ( u ( t ) )

+

b L ( u ( t ) )

A system is said to be time-invariant if its response to any arbitrary input signal does not depend on absolute time, i.e.:

y(t

+

a ) = L ( u ( t

+

a ) ) (2.4) So, if the normalized step responses to the different amplitudes obey equation 2.3, the investigated system is linear. If they also obey equation 2.4, the system is linear and time-invariant, in short: LTI. The different time instants at which the steps are applied are collected in the vector T = 120, 30, 401 ms, and the different amplitudes are collected in the vector

AF

= [20, 301 N. The amplitudes of the step are chosen in a range up to 1% of the passive belt force F,(t). The time values T are chosen in the time span in which the belt-system is tensioned (7-

>

18ms). For a proper comparison of the step

responses, these should be normalized, according to:

With x i f ( t ) the step response to the step with amplitude AFi at time instant ~ j , and x,(t) the orig- inal chest deflection. Note that the normalized step responses are shifted in time towards zero. The normalized responses for the different amplitudes are plotted in figure 2.8. The response for different force amplitudes is almost the same, so the system is dominantly linear. The normalized responses for the steps at different time instants are plotted in figure 2.9.

(a) r = 20 ms (b) r = 30 ms ( c ) T = 40 ms

Figure 2.8: Normalized step responses to different amplitudes

(a) A F = 20N (b) A F = 30N

gxlbi

,.

168

Figure 2.9: Normalized step responses at different time instants

. 6 -

Time-invariancy cannot be concluded from the plots. It seems that the system has more damping as time increases, and that the final value of the step response is time dependent too. This effect is caused by the dummy motion. The upper torso of the dummy rotates during the crash, which influences the transfer. This results in different gains at different time instants. To further investigate the time- invariancy, simulations have been done at time instants r = [20, 22, . .

.

,

46,

481

ms for A F = 30 N. The result of these simulations is pictured in figure 2.10. The system is not entirely time-invariant, but all responses stay within a band of approximately 25% of the mean value.

Identification and model reduction

,.a

,-\,

/ __-- , 6 -

For a first impression, the time variancy is neglected. Extended approximate realization is applied to the average of the normalized step responses to A F = 30 N, as displayed in figure 2.10. To calculate the average, the same time span for each normalized step response is used. The step response ends at 80 ms, and the maximum value of 7 is 48 ms, so the first 32 ms of each normalized step response is

used.

,-m

,,-/ , -- g 5 -

/

-

., L - g 5 -

E

n ,,

","

. - y--2--- ,2 '

&/\

1 2 3 - = 2 1 / /, -

/"

'k

o k o b ~ o d r s ob2 o m s 0b3 0d35 !O 0do5 obq ad15 o b ~ odzs o h l 0d35

-

0 0.01 0.02 0.03

t-z [s]

Figure 2.10: Normalized step responses for r = [20,22,

. . .

,46,48] ms and A F = 30N The largest ten singular values of the transformed Hankel matrix, calculated according to equa- tion A.4, are displayed in figure 2.1 l, and in table 2.2. From this values, it seems justified to choose

p = 2, which results in a second order model.

Figure 2.11 : Largest ten singular values of T Table 2.2: Largest five singular values The second order model can be described with:

C I S

+

c2

H ( s ) =

s2

+

2<w0s

+

w,2

The approximate realization procedure has resulted in an extra zero s =

2.

This zero causes the sys- tem to be non-minimum phase, which may cause problems with the controller design. The frequency of the zero is very high (approximately 8 x w,), so it has very little influence on the system behaviour. Therefore, this zero can be removed safely. Now, the general form is reduced to equation 2.7:

To check the validity of the model H ( s ) , its step response is compared with the averaged step response

the mass-spring-damper model of section 2.1. As can be seen, the values of both methods are close, indicating that using a second order model is justified.

Figure 2.12: Compare normalized step responses

Table 2.3: Parameters for the approximate realization of the mean step response, and for the mass- spring-damper model

Time variancy

Unit

radls

m/N

To investigate the influence of the time invariancy on the dynamic behaviour, approximate realization is applied to each individual step response at T = [20,22, . . . ,46,48] ms for A F = 30 N. The model

order is chosen second order. The Bode plot and Nyquist plot of each model is displayed in figure 2.13. As can be seen, the time-variancy has influence on the model. The eigenfrequencies vary from 190 to

Value msd-model 0.63 316 6.7 lo-' Parameter

I

W O c

(a) Bode plot (b) Nyquist plot (c) W O and

C

Value appr. real.

0.68 289 5.6 lou6

Figure 2.13: Comparison of models at r = [20,22, . .

.

,46,48]ms 13

320 rad/s. The dimensionless damping

5

varies from 0.2 to 0.8, with one outlier. This leads to a large gain difference at the eigenfrequency. However, for frequencies under and above the eigenfrequency, the error is only

f

3 dB in the Bode magnitude plot.

2.3 Input-output estimation via stepwise inputs on model

In 2.2.2, the model is estimated around the original passive belt force trajectory. However, the original and the controlled trajectory of F ( t ) can differ very much. Therefore, in this section estimation based on applying stepwise inputs will be performed. An advantage of using this method instead of estirnat- ing a model around the original belt force, is that it is easier to use, since it is not necessary to obtain the original belt force from the MADYMO model with the passive restraint system. The steps have magnitudes varying from 4 to 10 kN, and are not superposed to the original belt force Fo. Simulations have been performed for r =

[lo,

20, 251 ms and for AF = [4000, 5000, 6000, 7500, 10000] N. The normalized step responses are plotted in figure 2.14. Based on the singular values, figure 2.15, a sec- ond order model is considered to be appropriate. The Nyquist and Bode plots are shown in figure 2.16. The eigenfrequency and damping for each xi,j are shown in this figure too.

Figure 2.14: Normalized step responses to step- Figure 2.15: Singular values wise force inputs

With this approach, the average eigenfrequency for all plants is about 350 radls. The damping varies rather much, from 0.55 to 1. It decreases with increasing force, and with decreasing ri. There- fore, it is difficult to point out what the damping of the model is. When all step responses are averaged again, and approximate realization is applied to this average,

5

becomes 0.92, and w, equals 353 rad/s. This is slightly higher than the methods discussed before.

The step response of the model can be observed over a long time horizon. This gives a good insight in the time-variant behaviour. To show this, in figure 2.17, the step response to A F = 5000 N

at T = 10 ms is displayed. In this figure, different operating points can be distinguished. Roughly, four plateaus can be distinguished, between which a transition takes place, see table 2.4. These operation points are most probably caused by the dummy motion. During the crash, the dummy rotates, which influences the gain of the transfer. The dummy motion is most likely to be the cause of the time- dependent damping too.

(a) Nyquist plot (b) Bode plot (c) Eigenfrequency and damp- ing

Figure 2.16: Investigation of linearity and time-variancy

Figure 2.17: Normalized step response to

AF

=

5000 N at r = 10 ms

Time Value

0.04

5

t

5

0.04 6.2e-6

Table 2.4: Operating points

2.4

Conclusions and discussion

In this chapter, two methods have been discussed to obtain a model. The first one, using a mass- spring-damper model, provides analytical insight in the transfer from belt force to chest deflection. The second method, input-output estimation, provides insight using a numerical method to obtain a model. This latter method can be used for estimation based on stepwise perturbations to the original belt force trajectory, and estimation based on stepwise inputs to the model. The results of these two approaches are not very different, both result in a second order model, with similar parameters. The approach based on stepwise inputs to the MADYMO model results in higher damping and eigenfrequencies than the perturbation method. Applying stepwise inputs shows several operating points of the MADYMO model, which differ in gain and damping. This time variancy is neglected, and accounted for by using sufficient safety margins in the controller design. In the following chapters, the model obtained with the perturbation method has been used, since this has been proven successful in [2]. Control design based on the approach with stepwise inputs is a future research topic.

Chapter 3

Controlling the chest

deflection

3.1

Introduction

The control goal can be stated as:

Design a stablizing controller which manipulates the belt force, such that the chest deJEection follows the reference trajectory within the specijications.

3.1.1 Reference trajectory

The control action will be based on the tracking error. The reference trajectory should be designed such that the maximal chest deflection is minimal. Furthermore, the reference trajectory should satisfy the following constraints:

1. The occupant velocity at time t , has to be equal to the vehicle velocity, i.e.: xt(t,) = xieh(t,) 2. The occupant may not get in contact with the steering wheel, and not move backwards relative

to the vehicle, i.e.: 0 5

1

( x t - x,) - xVehl

I

lo Vt 't [to, t,]

The MADYMO model is too complex to determine a reference analytically. Therefore, the model of chapter 2 is used, and extended with a simple representation of the torso mass on the vehicle, figure 3.1. Also, the damper of model 2.2 has been removed, according to the observations made in section 2.1.4.

Figure 3.1 : mass-spring-damper model

So, the integral of x, is proportional to the velocity difference over

[to,

t,]. This velocity difference is a constant value, so the area under the curve of the reference trajectory for x, has to be constant. Therefore, to achieve a minimal maximum chest deflection, the optimal reference curve is a constant value. As it is very unrealistic to have a constant chest deflection at t = 0 seconds, it is slightly modified, by changing the reference to a sin2 function for t

<

t,. This provides a smooth transition at t = 0 and at t = t,, and avoids an infinite derivative of the reference trajectory, which can cause instability in MADYMO. t , is chosen to be 10 ms. Furthermore, if the dummy speed is equal to the vehicle speed, the force is reduced to zero. The crash can be considered as finished at this time instant

t,. After that, the reference trajectory doesn't matter anymore.

The value for the maximum can be calculated, since the area under the chest deflection has to be equal for different belt force trajectories. First, the area under the chest deflection of the model with the passive restraint system is computed. The chest deflection is pictured in figure 3.2. The area under the

- 0.0027

curve is 0.0027 m.s. Now, if

t ,

is known, the value for

x,,,,,

can be computed with x,,,,, - T.

Time [s]

0 0.05 0.1 0.15

Time [s]

Figure 3.2: Chest deflection with the passive re- Figure 3.3 : Reference trajectory straint system

Due to the many assumptions underlying this derivation of the reference trajectory, some iterations may be needed to assure that no contact between the dummy and the steering wheel occurs.

3.1.2

Control setup

The control setup is shown in figure 3.4, with the tracking error e = r - y, plant input u, and d the disturbance on the plant output y. In this chapter, two possible approaches to design a controller are discussed: loop-shaping, and robust control.

Figure 3.4: Control setup

For both approaches, the transfers important for controller design are defined as: 18

Open loop L = HC

Sensitivity S =

&,

which maps the reference r and the disturbance d to the error e: S =

E ( s ) -

Eo

R(s) - D ( s ) '

Complementary sensitivity T = &, which maps the reference r to the output y: T =

a.

R ( s )

Control sensitivity R = &, which maps the reference r and the disturbance d to the input ,

.

s

U ( s ) - U(")

R ( s ) D ( s ) '

00

In these equations, X ( s ) is the Laplace transform of the signal x ( t ) , defined as X ( s ) =

J

x(t)eFstdt.

0

3.2 Controller design with loop-shaping

3.2.1

Approach

In the loop-shaping approach, the sensitivity function S plays an important role, since it maps the reference trajectory r and the disturbance d to the error e. Besides that, the sensitivity can be related

to stability criteria. The general form of the sensitivity function is low gain for low frequencies, then rising to a certain peak value, and approaching 0 dB for high frequencies. The lower the low-frequent gain, the higher the peak value. This is known as the waterbed-effect. By specifying performance and stability requirements, boundaries are created for

S.

These performance criteria are specified in time domain, and then translated to the frequency domain, by observing the frequency content of the considered signals. The controller is designed in such a way, that S satisfies these boundaries, and thus satisfies the desired performance.

Performance specifications

To achieve a small tracking error e = r - y,

S

has to be small in the frequency range of r and d. The

maximum frequency of the reference trajectory r is 25 Hz due to the sin2 function. The characteristic

of d is unknown, but it can be seen as a model uncertainty. Generally, these disturbances are low-

frequent relative to the plant eigenfrequency of 46 Hz. Therefore, the maximum frequency of r is taken as the maximum frequency for which the controller has to achieve a desired performance. This performance is specified with the maximal tracking error em,,. The performance criterion for S then becomes S 5 e V f 5 25 Hz. The resulting boundary is pictured in figure 3.6 as line piece 1.

Stability criteria

The closed-loop transfer T =

&

is stable, as long as the loop gain of HC passes with the point

( - 1 , 0 ) on the left hand side, going from low to high frequencies, see figure 3.5. Two criteria that

indicate how far HC from (-1,O) - and thus instability - is, are the phase margin P M and the gain margin GM, pictured in figure 3.5.

With

IS/

=

l&l

= /distance . 1 to (-1,O)l' it turns out that the sensitivity is maximal, if the distance to (-1,O) is minimal. The sensitivity peak thus represents the frequency for which the loop

gain is closest to (-1, O ) , see figure 3.5. Therefore, S,,, can be linked to the stability margins as follows:

1 s m u x 5 -

Figure 3.5: Visualization phase and gain margin in the Nyquist plot

This means, that if we specify a GM and P M , the minimum of equations 3.2 and 3.3 constitutes an upper bound for S, pictured in figure 3.6 as line piece 3.

Finally, the bandwidth has to be determined. The bandwidth has to be high enough to satisfy the performance criteria. On the other hand, it should be kept as low as possible, because of two reasons:

0 A high bandwidth may cause problems in MADYMO, with respect to noise.

0 The method can be used in the future to design a controller which is applied in practice. An

actuator with a low bandwidth has a lower cost than one with high bandwidth.

To obtain an approximation for the desired bandwidth, we consider that at the bandwidth, the phase of

H C has to be at least -(l8O- P M ) . L H C can be approximated by - LS, since S =

&

FZ

&

for

H C

>>

1. A line is drawn with phase equal to (180 - P M ) , starting from performance specification 1 in the Bode plot of S. The desired bandwidth can be approximated by the 0 dB intersection of that line. For simplicity, we assumed that the bandwidth of S is an approximation for the bandwidth of H C .

This is not exactly true, because if IHC = 1, then IS =

#

1. But, since this is only meant as an approximation for the bandwidth, it needn't be very exact. The line is pictured in figure 3.6 as line piece 2.

3.2.2

Design

The maximum allowable error is defined as 5 % relative to the reference, which results in a low- frequency bound of -26 dB. Guidelines for the stability margins are proposed in [7], namely a phase margin of 30 O, and a gain margin of 2. The values are chosen a bit higher, since the uncertainty is quite large due to the neglected time invariancy. The phase margin is chosen as 45 O, and gain margin

is 2. According to equations 3.2 and 3.3, this sets the upperbound for S at 1.27, which is 2.1 dB. These two bounds, together with the line with phase 180 - 45 = 135O, give the 0 dB cross-over frequency

wbw at 185 Hz.

The most simple controller is a P-controller, which simply increases the gain of the open loop. The effect of a controller with increasing P is pictured in figure 3.7. P is chosen in such a way, that the

(a) Sensitivity S (b) Open loop HC

Figure 3.7: Effect of increasing P

bandwidth of HC exactly matches the desired bandwidth. This can be done by observing the plant gain, and then choosing P such that the open loop has the 0 dB cross-over at the desired bandwidth. The result of this P-controller is pictured in figure 3.8. As can be seen, a P-controller alone doesn't satisfy all performance criteria. The peak value is much too high, which means that the stability margins are too small. Furthermore, the low-frequent gain is too large.

A D-action is added, which increases the phase margin. With figure 3.5, it is clear that by adding phase, the distance to (-1,O) increases, and thus

S,,,

is lowered. The transfer function is given in equation 3.4. - 1 s + 1 T D , l D = 's+1 (3.4) r D , 2

The effect of increasing the D-action of a controller is pictured in figure 3.9. In the interval ~ 0

5

, ~

f

<

'TD.2, the controller adds phase. The greater the factor

z,

the greater the frequency range over which phase is added, and thus also the effect of D.

(a) Sensitivity S (b) Open loop HC (c) Nyquist plot

Figure 3.8: Effect of bandwidth-matching P-controller

(a) Sensitivity S (b) Open loop H C

Figure 3.9: Effect of increasing D

To lower the peak satisfactorily, T D , ~ is chosen at

9,

and at 4 wb,. Adding phase also shifts

the bandwidth to a higher frequency, because the slope of H C increases. To compensate the shift in bandwidth, P has to be lowered. This leads to an increase of the gain

IS/

at lower frequencies. The result of this PD-controller is pictured in figure 3.10. Bandwidth and peak value of the sensitivity are within the specifications, but the low-frequent gain is too high.

To lower the low-frequent gain, an I-controller is added, with transfer function 3.5.

The effect of an increasing I-action is pictured in figure 3.1 1. The I-action gives a higher controller gain for frequencies lower than 7-1. This leads to a lower gain for S at low frequencies, which means improved tracking behaviour at low frequencies. The I-action also gives a zero steady-state error, since T = s 1. These improvements have a cost, because by lowering the low-frequent

(a) Sensitivity S (b) Open loop H C ( c ) Nyquist plot

Figure 3.10: Effect of PD-controller

(a) Sensitivity S (b) Open loop H C

Figure 3.1 1 : Effect of increasing I

sensitivity gain, the peak value will be higher. For S not violating the high-frequent boundary, the cut-off frequency T I is placed at

2.

The result of this PID-controller is pictured in figure 3.12. The

resulting PID-controller, which achieves the desired

S,

is pictured in figure 3.13.

3.2.3 Results

The controller is evaluated in two closed loop configurations: one with the second order model, ob- tained with approximate realization in section 2.2.2, and one with the MADYMO-model. The refer- ence trajectory is adapted slightly such that contact between the dummy and the steering wheel doesn't occur. This results in a maximum chest deflection of 28 mrn. The results of the simplified model are shown in figure 3.14, together with the results of the MADYMO simulation. Note that the force is reduced to zero after t = t , = 100 ms, because then the dummy speed is equal to the vehicle speed, and the crash can be considered as finished. The simplified model predicts the MADYMO results very good. Apparently, the simplified model, which was estimated locally, is valid globally. The force from the MADYMO-model is quite different. This is due to the operating points, observed earlier in section 2.3. Apparently, the controller is robust enough to deal with these changes. It cannot be proved

(a) Sensitivity S (b) Open loop HC (c) Nyquist plot

Figure 3.12: Effect of PID-controller

Figure 3.13: PID-controller resulting from loop-shaping

mathematically, but the closed loop with MADYMO seems stable. Further, the tracking error exceeds the specifications during the first 6 ms, and the controlled system has a little overshoot.

-5

0 0 05 0 1 0 1

Tme [s]

(a) Belt force (b) Chest deflection (c) Tracking error

Figure 3.14: Comparison MADYMO and 2nd order model

A comparison with the original situation is done in figure 3.15. As can be seen, the controlled situation achieves a much better maximum chest deflection, approximately 43 % lower. It has to be

noted that in the original situation the airbag is not removed. The airbag places an extra force on the chest, so the beltforce is much lower.

Time [s]

(a) Belt force (b) Chest deflection

Figure 3.15: Comparison controlled and original model

3.2.4

Robustness

The controller was designed for a model based on the averaged step response. It is interesting to investigate whether the controller still functions properly if it is applied to a model, obtained from a step response other than the average. The models Hz which are estimated based on the step response at the time instants 7% = [20, 22, . . .46,48], are available. The controller will be tested on the model

that deviates the most from the average model. To determine which model deviates the most, an error criterion is needed. We calculate for each frequency the absolute error E =

/

H z ( j w ) - H,,(jw)

1.

Then,

the error criterion is defined as the integral of c over the frequency range:

This error criterion is plotted for each model Hi in figure 3.16. With this criterion, it follows that model

Figure 3.16: Error criterion

Hzo deviates the most. The sensitivity S, open loop transfer H C , and the Nyquist plot of H C with this model are shown in figure 3.18. From the Nyquist plot, it can be concluded that the influence of the model error on the stability margins is very strong. These margins are much lower, but the system is still stable. The results of a closed loop simulation with this model are pictured in figure 3.18. The closed loop is still stable, but the performance has detoriated.

-60

_

-80

o ' lo\lo' $0' 10' ro'

-200

ro" lo" -0' loi ro' lo'

ru [Hz,

(a) Resulting sensitivity S (b) Resulting open loop HC (c) Nyquist plot

Figure 3.17: Effect of changing plant

-11 006

4 008

n o r

0 05 0 1

Time 1%) 0 15

(a) Belt force (b) Chest deflection (c) Tracking error

Figure 3.18 : Simulation with deviating model

3.3 Controller design with robust control

In the previous section, the Nyquist plot is used to test the closed loop stability and some margins were taken into account to stay far enough from instability. It was readily observed that as soon as the loop gain passes the point 1 too close, the closed loop system becomes 'nervous'. It is then in a kind of transition phase towards actual instability. And, if the dynamics of the controlled process deviate somewhat from the nominal model, the shift may cause the encirclement of the point 1 resulting in an unstable system. The proposed margins were really rules of thumb: the allowed perturbations in dynamics were not quantized and only stability of the closed loop is guarded, not the performance. Robust control tries to overcome these deficiencies i.e. provide very strict and well defined criteria, define clear descriptions and bounds for the allowed perturbations and not only guarantee robustness for stability but also for the total performance of the closed loop system. Consequently a definition of robust control could be stated as:

Design a controller such that some level ofpegormance of the controlled system is guaranteed irrespective of model uncertainties.

The added value of robust control for this project is, that it is easily extendable to multivariable control, which is useful when more injury criteria are considered. Further, constraints can be taken into account by the filter design used in robust theory.

Model uncertainty

The model uncertainty

A H

is based on the observation that all step responses are within a bound of approximately 25% around the averaged step response2. Therefore, the model uncertainty

A H

is taken as 0.25H. The bode diagram of the uncertainty is given in figure 3.19.

Figure 3.19:

A H

3.3.2 Approach

An extensive treatment of robust theory is given in [8]. Here, only a short outline will be given. Important for the design of a controller for our model are:

The sensitivity S is important for disturbance attenuation and tracking performance, and is bounded by stability and performance criteria.

0 The control sensitivity R is important for stability robustness, and is bounded by the model

uncertainty.

Again, performance criteria will be defined. Next, filters are designed for each signal in the closed loop. With these filters, frequency dependent bounds for S and R are defined, analogous to section 3.2. The block-scheme of the closed loop with these filters is pictured in figure 3.20.

There are two classes of filters: characterization filters

V,

and penalizing filters W,. The input of a characterizing filter is an exogenous3 signal s:, which is normalized to 1, and the output is the signal s. The input of a penalizing filter is a signal s, and the output is an exogenous signal s:, which is normalized to 1. So, with the new closed loop configuration, the exogenous inputs and outputs are all normalized to 1. Now, if it can be guaranteed that the maximum of every transfer between exogenous signals st is smaller than 1, the system is stable. If we think an extra block

A

between signals u'

and dt, the model uncertainty can be characterized with

A H

=

VdAWu.

To reduce the number of

2 h o t h e r measure for the model uncertainty may be based on the Nyquist plots of all plants H z , as done in section 3.2.4.

' ~ n exogenous signal is a signal which is unknown, but bounded.

Figure 3.20: Filters in control structure Figure 3.21 : Augmented plant

exogenous inputs, r' and d' are taken together in one signal n'. This is possible, because r' and d'

effectively enter at the same point in the closed loop. For nf we can write: n,,, = r,,,

+

dm,,.

Figure 3.20 is redrawn by grouping the exogenous inputs on the left, and the exogenous outputs on the right side. This is pictured in figure 3.21. Now, the relation between the input and the output signals can be written as:

u 0

-H

And, with u = Ce, the relation between the exogenous signals can be written as:

This is known as the 'mixed sensitivity problem', [8]. According to the theory, a so-called Em con- troller is calculated, which minimizes

11

M ( C ) / ] ,4. If the controller achieves

//

M ( C )

II,<

y

=

1,

this implies:

For robust stability, it is necessary that

11

R A H

ll,<

1, as is shown in appendix B. Using equa- tion 3.9, it follows that:

So, we need to make sure that the product of the filters

I

WuVn

to guarantee robust stability. Summarizing, the bounds are:

Control sensitivity: IRl 5

&

Sensitivity: IS1 5

&

IW

v

I

Uncertainty:

1

A H ] 5

+-

I

is greater than the uncertainty

4/1

H 11-rs u p l H ( j w ) I , where 'sup' stands for supremum, which effectively indicates the maximum.

3.3.3

Filter design

The filters have to characterize and penalize the signals in the system, and by doing so, put bounds on S and R. Note that it is more convenient to put bounds on S and T, since S and T are related according to S

+

T = I. Therefore, in the following T is used instead of

R,

and calculated by T = HR. The filters are chosen in the following order:

1. Choose Vn so that it describes the reference and disturbance frequency characteristic. 2. Choose W, so that the product of (W,Vn( is above

/AH/,

according to equation 3.10.

3. Choose We based on the performance requirements, and make sure that the crossing of S and

T is above 0 dB [SI.

Characterizing filter Vn

The only characterizing filter is Vn, which characterizes n , representing the disturbance and reference. Typically, this is a low-pass filter, due to the low-frequent character of r and d. The amplitude of the signal n is defined as r,,, -k dm,, = r,,,

+

0.25rm,,. The cut-off frequency is defined just above the plant eigenfrequency, assuming that all disturbances resulting from plant uncertainty are taken into account. For higher frequencies, the amplitude is expected to be very low, so a 2nd order low-pass filter is chosen. This filter is described by equation 3.1 1.

With: n,,, = 1 , 2 5

.

28 = 35 mm, f , , ~ = 60 HZ and fn,2 = 1000 Hz. The Bode plot of this filter is given in figure 3.22.

Figure 3.22: Bode plot of Vn

Penalizing filter W ,

W, is a penalty filter for the plant input u. Typically, this has a high-pass characteristic, since high- frequent plant inputs are not desired. The amplitude of W , is chosen, such that lW,Vnj

>

1

A H 1 The form of this filter is first order high-pass, described by equation 3.12 and pictured in figure 3.23.

1 f u z

*

W" =

G(G)

s + f U J (3.12)

Figure 3.23: Bode plot of W ,

Penalizing filter We

Finally, We is chosen to obtain the desired performance. A limitation for We is the fact that the

crossing of S and T has to be above 0 dB, since S

+

T = I. When the earlier defined performance specifications are used, is the crossing under 0 dB. Therefore, the specification for the maximal error is relaxed to 5 mm. Further, an integrating action is forced in the controller by specifying a pole near zero. Then, the filter becomes a second order filter, described by equation 3.13, and pictured in figure 3.24.

1 fn 2 s2+(fe,l+fe,3)s+fe,lfe,3

We = s2+(fe,z+fe,o)s+fe,2fe,o (3.13)

With: em,, = 5 mm, fe,o = 1

.

lop5

Hz, f , , ~ = 2 Hz, f~ = 25 Hz and fe,s = 1000 Hz. These filters

Figure 3.24: Bode plot of We

result in the bounds for S and T, as pictured in figure 3.25 (a). The product of

I

W,V,/ lies above A H ,

as can be seen in figure 3.25 (b).

3.3.4 Results

The robust algorithm results in a 7th order controller which accomplishes y = 0.8856. The filters can be changed, such that y

=

1, but this is not done here, since this is only meant to show how to design a robust controller, and to show the performance. The resulting controller and the resulting S and T are pictured in figure 3.26. Because the controller order is very high, simulations take very long, up to 40 hours. Further, it is not very clear what the structure of the controller is. Therefore, the controller order is reduced by investigating the singular values of the balanced realization5 of the controller. These are given in table 3.1. From these values, it can be concluded that reducing to a PID-controller, which is

'Inthe balanced realization, the states are ordered corresponding to their relevance for the input-output behaviour, see [9].

(a) Bounds for S and T (b)

I

W,V,/ above /AH1

Figure 3.25: Results filter design

(a) Resulting controller (b) Resulting S and T

Figure 3.26: Bounds and the resulting S and T

Table 3.1: Singular values of balanced realization robust controller

third order, is justified. The effect of reducing the order is pictured in figure 3.26 (a), as the dashed line. Roughly, this controller is a PID-controller, with an I-action until 2 Hz, and from 100 Hz a D-

action. The Nyquist plot of the open loop, and the S and T resulting from the reduced controller are compared with the full-order controller in figure 3.27. The stability margins of the reduced controller are smaller, and both S and T have a higher peak. This means that robust stability with this model

(a) Nyquist plot HC (b) S and T

Figure 3.27: Effect of reduced order controller error is not guaranteed anymore.

Simulation results with MADYMO and the simplified model are pictured in figure 3.28. As can be seen, the error is within the specifications. Due to the trade-off between stability and performance, the reference trajectory is not followed very well, which leads to contact of the upper torso with the steering wheel at t = 93 ms. A better design of the filters can prevent this from happening.

Time [s]

-5 1 I

0 0.05 0.1 0.15

Time [s]

(a) Chest deflection (b) Belt force (c) Tracking error

Figure 3.28: Simulation results MADYMO with reduced order controller

3.3.5 Conclusions and discussion

With robust control design filters are designed, which characterize and penalize the signals in the closed loop. These filters put bounds on the relevant transfers S and R. The resulting controller guar- antees robust stability for the model error AH. Advantages of this method are:

Guaranteed stability for defined model uncertainty.

e The method can easily be extended to multivariable approach, to consider more injury criteria. By using penalizing filters for the signals, constraints can be taken into account.

A disadvantage is the higher order of the resulting controller, since this is equal to the order of the plant and the filters together. Due to the high order, simulations take very long, and the controller structure is not very clear anymore. Here, this problem is tackled by reducing the controller order, but this doesn't guarantee the robustness of the closed loop. Algorithms are available which give the robust controller a fixed structure, e.g. PID [lo]. This has to be researched further.

Chapter 4

Discussion

4.1

Combining belt and airbag

So far, only the belt is used to restrain the dummy. In practice, one or more airbags are also used to decelerate the dummy. The effect of using an airbag is investigated with a MADYMO simulation. The airbag is considered as a disturbing force, so the controller has to be robust enough to deal with this. The results of a simulation with airbag are shown in figure 4.2. Since the airbag causes an extra

Time [s]

(a) Belt force

Time [s]

(b) Chest deflection

Figure 4.1 : Simulation results with airbag.

chest deflection, the belt force is gradually reduced to zero. When the force reaches zero, a numerical instability in MADYMO causes the force to reach very high values. Most probably, this instability is caused by the transition from a positive to a negative force. On the moment of transition, the belt becomes an undetermined system. A solution to this problem may be forcing the belt force to stay zero when it reaches zero. In this simulation, the force is reduced to zero after 100 ms, which was the original setting. If the results are considered until the moment of instability, it can be concluded that the controller is robust enough to deal with the disturbance caused by the airbag. This means that it might be possible to use separate controllers for the belt and airbag. Controlling both airbag and belt force simultaneously is subject of further research.

4.2 Chest deflection versus chest acceleration

The goal stated in chapter 1 was to minimize the chest deflection. Reducing the chest deflection, however, has an effect on other injury criteria. For example, another injury criterion is the chest ac- celeration. Minimizing this criterion according to the same strategy as used in this report is treated in [2]. It is interesting too see how minimizing one criterion influences the other criterion.

First, a MADYMO simulation is done in which the chest deflection is controlled. Then, a sim- ulation with controlled chest acceleration is done. Results of these two simulations are shown in figure 4.2.

(a) Belt force (b) Chest deflection (c) Acceleration

Figure 4.2: Comparison controlled chest deflection respectively chest acceleration

When the chest acceleration is minimized, the chest deflection reaches quite high values: 0.04 m instead of 0.028 m, which is 43 % higher. If, on the other hand, the deflection is minimized, it turns out that the acceleration peak value is just 250 d s 2 instead of 200 d s 2 , which is only 25 %

higher. It seems better to control the chest deflection than the chest acceleration. A disadvantage of controlling the chest deflection is that it has no direct link to the dummy position. Better results for both criteria may be obtained by a controller which takes both injuries into account. This is subject of further research. Using robust control theory may be convenient, as it can easily be extended to a multivariable approach.

Chapter 5

Conclusions and recommendations

In this report, a method to reduce the maximal chest deflection in a USNCAP crash is discussed, by manipulating the belt force. First, two approaches to obtain a model are proposed. To obtain analyt- ical insight, a mass-spring-damper system is used. For a more quantitative approach, input-output estimation based on step responses is done. Both methods result in a second order, linear and time invariant model. Further, from the control results, it follows that the locally estimated model is valid globally. Next, a reference trajectory is designed which reflects a minimal maximum chest deflection. This trajectory is changed to fulfill realistic constraints. Then, a controller is designed which achieves tracking within predefined specifications. The controller is designed in two ways: by loop-shaping the sensitivity function, and with robust control. Evaluation in the MADYMO model gives good results for both methods. The robust controller has lower performance, but is more robust to changes and uncertainties in the dynamics of the transfer from belt force to chest deflection.

This report only discusses the control of the chest deflection by manipulating the belt force in one crash. Further research has to focus on the following items:

Multivariable control of more injuria criteria, such as head or chest acceleration. More research is needed in the control of both belt and airbag.

Real actuators pose constraints on the belt force. These constraints have to be taken into account, for example by using another control strategy, such as model predictive control (MPC).

Bibliography

[I] MADYMO manuals, TNO-Road Vehicles Research Institute, Delft, 1997.

[2] R.J. Hesseling, F.E. Veldpaus, M. Steinbuch, T. Klisch, Control Design for Safety Restraint Sys-

tems, in Proc. 3rd IEEE Int. Conf. on Control Theory and Applications; Editors: Jianliang Wang,

Pretoria, South Africa, 440-444, 2001.

[3] B. De Schutter, Minimal state-space realizatoin 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.331-354, Sept. 2000.

[4] J.B. van Helmont, A.J.J. van der Weiden, and H. Anneveld, Design of optimal controllers for a coalJired Benson boiler based on a modiJied approximate realization algorithm, in Application

of Multivariable System Techniques, London, pp.313-320, Elsevier, 1990.

[5] B.L. Ho An effective construction of realizations from input/output descriptions, Ph-D thesis, Stanford University, Stanford, California, 1966.

[6] S.Y. Kung A new identijication and model reduction algorithm via singular value composition,

in Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, California, pp. 705-714, 1978.

[7] G.F. Franklin, J.D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-

Wesley Publishing Company, 1994.

[XI A.A.H. Damen, S. Weiland, Robust Control, Course material, Eindhoven University of Technol-

ogy, Eindhoven, 200 1.

[9] R.J. Ober, Balanced realizations: canonical form, parametrization, model reduction, Int. J. Con-

trol, 46, 643-670, 1987.

[lo] V. Kapila, Fixed-Structure Robust Controller Synthesis for Systems with Elemental Uncertainty

Appendix A

Extended approximate realization

A measured step response can be written as:

With

Si

the value of the step response at time instant t i , and 0

<

ti

5

N A t . Now, the method of approximate realization for step responses can be applied to obtain a model, according to [4]. First, the matrix T is defined as:

with r

+

r' = N

+

1. Then, the following matrices are defined:

With the concept of singular value decomposition, the effective rank p of T can be determined, [6],

according to:

where the eigenvector matrix

[UI

Uz]

and [1/1 are orthogonal and the matrices C1 and C2 are semi-positive definite and diagonal. The singular values are a measure for the importance of the states for the input-output behaviour. A value for p is chosen, such that the matrix C1 consist of p singular values on the diagonal, with decreasing magnitude, so a1

>

a2

>

. . .

>

ap. Singular values ai with

i

>

p are significantly smaller than gp and are therefore neglected. Now, the system matrices A, B, C

of the model H, equation A.5 can be computed according to equation A.6:

The order of the estimated model H is equal to p, so model reduction is incorporated in the estimation procedure.

Appendix

B

Small gain theorem

Model uncertainties can have a significant influence on the closed loop behaviour. Therefore, an ad- ditive model uncertainty block is added to the control setup of figure 3.4, as pictured in figure B.l (a). This picture is equivalent to figure B.l (b), where the feedback loop is replaced by the equivalent transfer

R

=

&.

TO guarantee that the closed loop with model uncertainty is stable, we can apply

(a) Model uncertainty (b) Baby small gain theorem

Figure B. 1 : Model uncertainty in the closed loop

the so-called 'baby small gain theorem'. This states that if it is guaranteed that the amplitude of the transfer R A H is less than 1 for all frequencies, the Nyquist plot cannot encompass the point (-1, O ) , and stability is guaranteed. This is pictured in figure B.2. We can write this condition as:

Figure B.2: Small gain stability in Nyquist plot

Now, if we can characterize the model error AH, and guarantee that

11

A H

I

I

,

I

$,

a sufficient condition for stability is:

11

R

I

I

,

I

7. So, the model uncertainty puts a bound on R. The greater the uncertainty, the lower

11

R

11,

has to be to guarantee stability.

Appendix C

function [SYSTEM-C, Y-MR, S-diag] =hankelstepfun (y,rho)

% [SYSTEM-C, Y-MR, S-diag]=hankelstepfun (y, rho)

% computes approximate realization with order rho of step response y

% based on method described in [J.B. van Helmont]

n=length (y) ;

t= [O : le-4 : (n-1) *le-41' ;

3 3 _{o o S-E}

tmp-S-pl=hankel (y (2 :end) ) -y (1 :end-1) *ones (1, n-1) ;

tmp-S-p2=floor (size (tmptmp_S_p2=flooro/2);SSp11 1) /2) ;

S~p=tmp~S~pl(l:tmp~SS_p=tmp_S_plo;p2I1:tmp~S~p2); 9 0 3 0 S-A

S-A=S-p (2 : end, 1 : end-1) ;

%% singular value decomposition of H-E

[U, S,vl=svd (S-p) ;

% % diagonalize singular values to vector

S-diag=diag (S) ;

%% use only relevant partial matrices

U-l=U ( :

,

1 : rho) ;

V-l=V ( : ,I: rho) ;

%% construct approximate realization

A-S = sqrt (Sigma-1)- (-1)

*

U-1'

*

S-A

*

V-1

*

sqrt (Sigma-1) -(-1);

B-S = sqrt(Sigma-l)^(-1)

*

U-1'

*

S-B;

C-S = S-C

*

V-1

*

sqrt (Sigma-l)^(-1);

D-S = zeros (size (C-S, 1)

,

size (BPSI 2) ) ;

% % create discrete state space model

SYSTEM-D=ss (A-S, BPSI CCSI D-S, le-4) ;

% % generate stepresponse

[Y-MR, T-MR] =step SYD,ST(EMo;pYeM_ITMt_R=sR,] t) ;

%% error criterion

epsilon=abs (y-Y-MR)

.

/abs (y) ;

epsilon=cumsum(epsilon); epsilon=epsilon(end);

%% figure with step response

figure; hold on

plot (t, y, T-MR, Y-MR,

'

r--' ) ;

xlabel (It

[?I

'

,

' fontsize', 20)

ylabel ( I y (t) [ ? I

' ,

'

fontsize', 20)

legend('measured responser,'unit stepresp of H-{COM)')

tmp-title={'\epsilon =' epsilon);

title (tmp-title, ' fontsize'

,

10)

% try to convert to continuous plant

t rY

SYSTEMPC=d2c (SYSTEMsysTEM_c=d2cO;DI 'matchedr ) ;

disp('zp-matched d2c conversion') catch

SYSTEM_C=d2c (SYSTEM-D) ;

disp ( ' zp-not-matched d2c conversionr )