Black-box modeling of a semi-active motorcycle damper

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Black-box modeling of a semi-active

motorcycle damper

JOHN S ¨

ODERBERG

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Abstract

Different driving conditions demands different suspension settings, developing suspension systems often leads to a compromise between comfort, handling and driving character-istics. This problem, and the need to be able to implement more advanced suspension features led to the development of active and semi-active suspension systems.

Suspension system manufacturer ¨Ohlins Racing AB makes a semi-active system called CES, Continuously Controlled Electronic Suspension where the damping characteristics are controlled by a patented electronic CES valve. This master thesis project was initiated to create a black-box model of the CES system to be used for future control design improvement.

When creating a black-box model the main concerns are experiment design and the modeling procedure. This thesis proposes an experiment design that emulates the desired behavior of the CES system in order to generate measurement data for a control relevant model. The modeling procedure is performed using three different model structures: Polynomial NARX, Sigmoid network NARX and Neuro-Fuzzy model structure.

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Sammanfattning

Olika körsituationer kräver olika stötdämparinställningar, utvecklingen av stötdämpare leder därför ofta till en kompromiss mellan komfort, köregenskaper och vägh˚allning. Denna kompromiss, samt de ökande kraven p˚a mer avancerade stötdämparfunktioner har lett till utvecklingen av aktiva och semi-aktiva stötdämpare.

Stötdämpartillverkaren Öhlins Racing AB säljer ett semi-aktivt system kallat CES, Continuously Controlled Electronic Suspension där stötdämparkaraktäristiken regleras av en patenterad elektronisk CES-ventil. Detta examensarbete grundades i en önskan att skapa en black box-modell av en CES-stötdämpare för att i framtiden kunna förbättra regleringen av denna.

Det viktigaste när man skapar black box-modeller är designen av experimentet och valet av modellstrukturen. Detta examensarbete föresl˚ar en experimentdesign skapad för att efterlikna de önskade arbetssituationerna för CES-stötdämparen och därigenom generera mätdata som är relevant för en modell som ska användas till reglering. Mod-ellering sker sedan med tre olika modellstrukturer: Polynom-NARX, Sigmoid-NARX och Neuro-Fuzzy-struktur.

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Chapter 1 Introduction

A conventional hydraulic damper for motor vehicle suspension needs to be designed to work under a lot of different conditions. Different driving situations demands different damping characteristics. Fast cornering and good handling needs a hard and stiff damp-ing while a soft dampdamp-ing is needed for a smooth and comfortable ride. In conventional suspension systems this leads to a compromise between these conflicting conditions. A sports car usually handles well, corner with minimum roll and stop with minimum brake dive allowing it to keep its traction but the hard ride makes it uncomfortable for longer drives and bad roads. A luxury car however is soft in order to offer a smooth ride but ends up suffering from bad handling. To solve this problem the vehicle industry has been developing different kinds of active suspension systems over the years. The idea is that the damping characteristics should be continuously adjustable to better suit the current situation. A fully active damper can both add and remove energy from the system and therefore offer more possibilities in the control of the vehicles behavior. The disadvan-tages are however huge energy consumption and a higher level of complexity. A popular solution is therefore to use a semi-active damper in which energy only can be absorbed, in an energy efficient way. This is usually a traditional hydraulic damper with some kind of system adjusting the flow dynamics. This master thesis project aims at investigating dif-ferent ways of modeling a specific kind of semi-active damper using black-box techniques in order to get a better understanding of its behavior and enhance the performance of its control system.

1.1 Background

¨

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This is due to a lot of factors like the for example compressibility of the hydraulic fluid, valves and backlashes.

The CES valve is a big success and has recently sold over one million units to car manufacturers like Audi, Volvo, Ford, Volkswagen and Mercedes-Benz. But since ¨Ohlins Racings main focus is on motorcycle suspension systems they are developing the CES system for motorcycles as well. On a motorcycle however, the suspension system can be made more sophisticated than most automotive systems. This is mainly because the dynamics of a motorcycle are more complicated. Another reason is that the ratio between the sprung and unsprung mass is a lot smaller than on a car making it harder to obtain high riding comfort [1]. This makes the performance of the CES control system even more significant.

The damping force can easily be measured in a laboratory test rig but when the damper is in operation on a motorcycle it becomes difficult and expensive. Since most of the theory regarding automatic control assume available feedback of the output signal, this makes the control problem even more intricate. A possible way of obtaining this feedback is to use a mathematical model of the damper acting as an observer that calculates an estimate of the damping force. ¨Ohlins Racing are therefore interested in modeling the CES damper in order to increase the performance of the control system.

1.2 Objective

The objective of this master thesis project is to investigate the possibility of identifying a black-box model to use as an observer in a future control system for a CES damper. Some different model structures will be tested and compared against each other to find the one that best suits the CES damper. The effect of different kinds of input data to these model structures will also be studied.

1.3 Past work

In the autumn of 2009 a preceding master thesis project was initiated [7] where a phe-nomenological model of a CES damper was created in MATLAB/Simulink. The model was discovered to be too complex for execution in a real time control system. Because of this, the author Rasmus Loman, attempted to create a black box model of the system. Although a complete model was not made, his work indicated that it might be possible to successfully create a black box model (or a collection of sub models) useful for control purposes.

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Chapter 2 The CES damper

Figure 2.1: The CES damper

2.1 Semi-active suspension

The characteristics of a hydraulic damper can be defined by its speed-force plot. A typical speed-force plot is shown in Figure 2.2. The speed axis represents the speed of the piston rod, i.e. the speed with which the damper is compressed or retracted. As seen in this case, the higher the speed is the higher the damping force. As mentioned in Chapter 1, the result is often a compromise between comfort and handling. The need to achieve both has led to the development of active and semi-active suspension systems. The idea is that the damping characteristics should be continuously adjustable while driving so that the vehicle always performs optimally. A basic system could perhaps detect the current driving conditions and switch between different pre-programmed speed-force curves while a more advanced system could for example be able to detect individual bumps in the road and momentarily reduce the damping force.

2.2 A hydraulic damper

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Piston rod speed [m/s]

Damping force [kN]

Figure 2.2: A typical speed-force plot

cylinder, oil has to be evacuated via (D) through the CES valve (E) and into the reservoir (F). When the damper is rebounded the opposite occurs, but instead of the piston valve it is now the top valve (H) that is open. Now oil is flowing to the compression chamber (A) from both the reservoir (F) and the rebound chamber (C) through the CES valve (E). In a high performance damper like this one, the whole system is put under pressure. The oil reservoir is divided into two parts separated by a dividing piston and the area (G) is filled with nitrogen gas. This is done to avoid the risk of cavitation, that vapor bubbles are created around the piston when it is moving at high speeds, much like what happens around a propeller in water.

2.3 The CES valve

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B

D

A

C

H

G

F

E

Figure 2.3: A diagram of the CES damper

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2.4 The control problem

As mentioned in Section 2.1 the goal of the semi-active damper is to be able to alter the speed-force characteristics while driving. This is done by altering the control current put on the CES valve. Since the control reference is a speed-force curve, the reference value for the damping force is dependent on the velocity of the piston rod at every instant. The velocity is available through a derived measurement of the piston rod position. The damping force is however difficult to measure when the damper is mounted on a vehicle, which is why a model of the CES damper is desired. A model could either be used as an observer in a control system to estimate the damping force, or be inverted and used as a feed forward controller. This project is focused on creating a model to be used as an observer. A diagram of the proposed control strategy is shown in Figure 2.5.

The CES damper is a complex system with hysteretic and nonlinear dynamics. Ex-amples of the dynamics are shown in Chapter 5.

F

ref

C

G

ˆ

G

+ Fref − v i F ˆ F

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Chapter 3 System identification

System identification is the theory of building dynamical mathematical models from mea-sured input and output data. In contrast to traditional phenomenological modeling, system identification demands none or little knowledge of what is happening inside the system. The challenge is instead the choice of model structure and obtaining relevant data. The process of identifying a system is in many ways iterative and demands some trial and error. First of all you have to design an experiment that excites the dynamics that you want to capture in your model. The data then has to be evaluated and pre-processed. You then choose your type of model with which endless possibilities of different configurations, sizes and orders are available. This is where most of the time is spent and this is often automated in an iterative process of its own as will be shown later. When the best model structure is determined an optimization procedure fits its parameters to the data. The model then has to be tested and evaluated, if it is not sufficient a new iteration is needed.

3.1 Black-box models

In opposite to semi-physical grey-box models, no knowledge of the actual physical phe-nomena happening in the system is needed when building a black-box model. It is however good to have some knowledge of the behavior of the system when choosing what kind of black-box model you are going to use. A good rule is to always try simple things first [5] and move on to more advanced model structures if needed. Because the CES damper has highly nonlinear dynamics, linear model structures are not sufficient as shown by earlier work [7]. Hence only nonlinear models will be considered. The ones used are Polynomial NARX, Sigmoid network NARX and Neuro-Fuzzy models.

3.2 Lab equipment

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be fed with arbitrary signal files as reference for the displacement and valve current which makes it possible to reproduce actual recorded driving conditions, frequency sweeps, steps and such. The program then logs the time, displacement, current and measured damping force to be analyzed later.

Hydraulic actuator Damper

Figure 3.1: The principle of the dynamometer rig

3.3 Experiment design

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inputs can be manipulated quite freely so the when issue is rather a question of how to design the inputs. The experiment has to excite the dynamics that the model should capture, and deciding on what is relevant for the application in question is not always straightforward. In this case, when the goal is a simulation model, one has to account for the whole operating range.

Some papers have been written on the matter of experiment design for MR dampers, i.e. [8] where a frequency modulated displacement signal along with a binary electrical signal is proposed. In [9] experiments are carried out with displacement signals created from low pass filtering white noise, with both constant and varying electrical current. The low pass filtered white noise can be seen as an attempt to create a road like signal. This goes hand in hand with a theory from [2] that data from the desired operating conditions of the model is very useful for the identification process. It is also mentioned in [2] that closed-loop identification may help to obtain control relevant models for LTI systems. Even though this is not a linear system, most of the experiments carried out within this project are done on a system with ¨Ohlins Racings present control solution connected. The idea is that the main focus should be on designing the displacement input and let the present control solution provide an electric current input signal. With this setup the experiment data, especially the correlation between the two inputs, becomes more like in the expected operating conditions focusing the modeling on the control relevant behavior. Before modeling, it is always a good idea to do pretreatment of the measurement data. This involves resampling, filtering, removing outliers and other preparations to make sure that the modeling process does not have to suffer from bad data quality. The data also has to be split into sections for estimation and validation and formatted to fit the modeling software.

3.4 NARX modeling

The inputs to these model structures are called regressors. A regressor is a variable containing old inputs or outputs from the system, for example y(t − 1) is the system’s output delayed one sample and u2(t − 3) is the system’s second input delayed by three samples. The choice of regressors, called the regression vector, is really important for the outcome of the modeling process and demands both experience and knowledge of the system. NARX (Nonlinear AutoRegressive model with eXogenous inputs) is a class of nonlinear models that in some sense origin from the well known ARX models common in system identification. As with ARX models, NARX models are discrete time input-output equations where the output y at time k depends on old values of itself and its input u up to the time k − 1.

y(k) = f (y(k − 1), . . . , y(k − ny), u(k − 1), . . . , u(k − nu)) (3.1) Where f is a nonlinear function deciding on the kind of NARX model structure. In this project f can be a polynomial or a sigmoid network and u represents multiple inputs u1, u2, . . . , um.

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set. This overfit is of no benefit to the model since the model will be subject to different noise realizations every time it is used.

3.4.1 Polynomial NARX models

If the nonlinear function f in equation (3.1) is a polynomial the resulting model structure is called a polynomial NARX model. The nonlinear regressors of the polynomial NARX model are products of powers of the earlier mentioned regressors. A nonlinear regressor could for example look like:

Reg = y(t − 2) · u1(t − 3) · u2(t − 2)2· u2(t − 3) (3.2) Where y(t − 2) is the systems output delayed by two samples and u1, u2 is the two input signals to the system. The full polynomial NARX model will be on the form:

y(t) = a0+ N X

i=1

ai· Regi (3.3)

Where ai is a weight. This kind of model structure makes computer implementation straightforward which makes them well suited for model-based control. In [3] it is shown that polynomial NARX models can exhibit a wide variety of hysteretic behaviors similar to those of the CES damper. To further enhance the model’s ability to describe com-plex systems, non-integer powers can be used in the nonlinear regressors and additional fabricated inputs may be added.

3.4.2 Sigmoid network NARX models

In this NARX model structure the nonlinear function f in equation (3.1) maps the regres-sors to the model output through a single layer sigmoid network. The sigmoid network has a structure like:

g(x) = n X

k=1

αkK(βk(x − γk)) (3.4)

Where K(s) = (es₊₁₎−1_{is the sigmoid function, β}

kis a row vector such that βk(x−γk) is a scalar and x is the regression vector. The number of sigmoid nodes in the network is decided by n. For more information see [5]. It is possible to use the nonlinear regressors from Section 3.4.1 as inputs to this model structure as well but it has not shown to be of any benefit in this project.

3.4.3 Estimating the parameters

The model parameters are estimated using the nlarx tool in System Identification Toolbox for MATLAB.

3.4.4 Automated regressor selection

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Initialize

- Measurement data - Candidate regressors - Initial model

Evaluate model

- Calculate cost function

Add random regressor

to the model and remove it from the candidate

regressor set Final model Stop? No Yes Evaluate model

- Calculate cost function

Better?

Remove latest regressor

from the model

No Yes

Prune

- Remove one regressor at a time - Evaluate the resulting models - If a better model is found, use that one instead and prune again

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Initialization

The algorithm has to be initialized, this is done by supplying the measurement data in an appropriate format, generating the set of candidate regressors and picking the initial model. The measurement data has to be in a format compatible with the software that is going to be used for parameter estimation and simulation. The set of candidate regressors is then generated by creating a list of the allowed outputs/inputs with their respective delays. This will do for the sigmoid NARX model structure but when this method is applied on the polynomial NARX models the candidate regressor set is a bit different. In that case the list of candidate regressors is used to create a new list of nonlinear regressors by combining the regressors with different delays and powers according to predetermined rules. These rules have to be restrictive as the amount of nonlinear regressors easily can grow out of proportion and make the process unnecessarily time consuming. Regardless of model structure, an initial guess has to be made in order to have something to improve. Model evaluation

In the model evaluation step the model parameters are estimated to fit the estimation data and the model is then simulated using the validation data. The result of the simulation is compared to the measured output from the system belonging to the validation data. The difference between the simulated and measured output results in a cost value. The cost function is borrowed from [5] and is shown below.

Cost J = ||ym− ysim|| ||ym− ¯y||

· 100 (3.5)

Here ym is the measured output, ysim is the simulated output and ¯y is the mean of the measured output. A low value means low cost and a good fit while the value 100 means that the model is nothing better than just using the mean as estimation, note that the value could get higher than 100. The cost value makes it straightforward to decide if a new model is better than the previous one.

Adding a random regressor

One new regressor is picked at random from the list of candidate regressors and removed from that list to avoid reusing it. The regressor is then added to the regression vector and the resulting model is evaluated. If the new regressor leads to a reduced cost it is accepted and the pruning begins, if not it is removed and a new regressor is picked. Pruning

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Stop criterion

The algorithm is stopped when the stop criterion is met. The most common stop criterion is that the candidate regressor list is empty, all allowed regressors have been tested. It could for example also be set to stop at a certain cost threshold or a specified execution time.

3.5 Neuro-Fuzzy modeling

Within this project some studies were made on neuro-fuzzy models. The Fuzzy Logic toolbox for MATLAB includes tools for working with ANFIS (Adaptive Neuro-Fuzzy Inference Systems). This is a kind of model structure that combines fuzzy logic with neural networks and is said to be able to emulate most nonlinear systems. The ANFIS tool uses a learning algorithm based on back propagation gradient descent and least squares methods to iteratively create a fuzzy inference system [10]. This thesis will not go into depth in the fuzzy logic theory, the interested reader is referred to literature on the subject. The ANFIS tool has an easy to use interface where the model structure parameters are tuned. The modeling procedure is mostly trial and error but the choice of model size is obviously a tradeoff between precision and computational cost. The ANFIS structure can be seen in Figure 3.3.

All the parameters in (1) are shown as follows

f

0

is the offset force of the MR damper, c

b

is the slope

coefficient of the hysteresis curve, f

_y

and k are two

coefficients characterizing the maximal damping force, and

c

w

is the width coefficient of the hysteresis curve,

respectively. Fig. 1 shows the relation for the input voltages

0V, 2V, 4V and 7.5V, respectively when f

0

=20, c

b

=1, f

y

=300,

k=1 and c

_w

=40.

III. A

DAPTIVE NEURO

-

FUZZY INFERENCE SYSTEM

A. TSK fuzzy model

TSK fuzzy model was put forward by Takagi, Sugeno

and Kang [7], which can determine the fuzz system by

adaptively generating the fuzzy rules based on input and

output data. The classic fuzzy rule of TSK fuzzy model is as

follows

If x is A and y is B

then z=f(x, y) .

Where x,y denote input language variable, A and B are the

fuzzy sets, while z=f(x,y) is the exact function in the

conclusion.

The inputs for the fuzzy model are fuzzy and the outputs

are exact, so that the total output of the fuzzy system is easy

to acquire only by weighted averaging without solving fuzzy

process. TSK fuzzy model is easy to parameterize and can be

changed into the adaptive neural network system which their

parameters are controllable. The TSK fuzzy model is also

called adaptive neuro-fuzzy inference system (ANFIS).

B. The first-order TSK fuzzy system and ANFIS structure

If a first-order TSK fuzzy system consists of two inputs

and one output, and there are two IF-THEN fuzzy rules as

follows

Rule 1: If x is A

1

and y is B

1

, then z

1

=p

1

x+q

1

y+r

1

Rule 2: If x is A

2

and y is B

2

, then z

2

=p

2

x+q

2

y+r

2

Figure 2. Inference process of the first-order TSK fuzzy system.

Where x,y are input language variables. A

1

,A

2

,B

1

and B

2

are

fuzzy sets. z

1

,z

2

are output language variables. p

1

,q

1

,r

1

,p

2

,q

2

and r

2

are the output parameters of the fuzzy system.

Fig. 2 shows the inference process of the fuzzy system,

where

_{W W}

₁

_,

₂

mean the fitness of the fuzzy rule,

W W

₁

,

₂

denote

the normalized fitness.

Figure 3. ANFIS structure of first-order TSK system.

Fig. 3 illustrates the equivalent ANFIS structure to the

TSK fuzzy system above. Where each layer denotes:

L1: solving the fuzzy membership of inputs.

L2: solving the fitness of each fuzzy rule.

L3: solving the normalized fitness.

L4: solving the output of each fuzzy rule.

L5: solving the overall output of the fuzzy system.

IV. T

HE NEURO

-

FUZZY SYSTEM OF THE

MR

DAMPER

A. ANFIS of the direct model

The physical model of MR damper consists of three

inputs and one output according to (1), the inputs are the

relative velocity, the relative acceleration and the control

voltage, the output is damping force. The ANFIS system of

the direct model of MR damper has the same inputs and

outputs with the physical model of MR damper (Fig. 4).

Figure 4. The sketch of the ANFIS of direct model of MR Damper.

Therefore, the first-order TSK fuzzy model consists of

three inputs and one output, the

ist IF-THEN fuzzy rule reads

Rule i If

u

is A

i

and

u

is B

i

and

V

is C

i

then, y

i

=p

i

u

+q

i

u

+t

i

V

+r

i

where A

_i

,B

_i

and C

_i

are fuzzy sets,

u

,

u

and

V

are input

language variables, y

_i

is output language variables, p

_i

,q

_i

,t

_i

and r

_i

are the output parameters of fuzzy system.

V

u

f

MR

Damper

ANFIS of

direct model

+

−

•

• ˆf

• W

2

W

1

A

1

A

2

B

1

B

2

z

1

=p

1

x+q

1

y+r

z

2

=p

2

x+q

2

y+r

1 1 2 2 1 1 2 2 1 2

W z

z

W z

W

+

=

+

x

y

X

Y

1.1( 2.3)

1

y yl V

f

e

− −

=

+

0

_{1 1.81}

0.2 w V

c

x

e

−

=

+

1.04

1 10.34

b b V

c

C

e

−

=

+

L1 x y N N

S

1 W 2 W z x y x y A1 A2 B1 B2 L2 L3 L4 L5 W1 W2 z1 z2

461

465

Figure 3.3: The first order ANFIS structure, picture borrowed from [11]

Where x, y are input variables, z the output. A1, A2, B1, B2 are fuzzy sets. The structure is divided into layers solving different parts of the system.

L1: calculates the fuzzy memberships of the inputs. L2: calculates the fitness of each fuzzy rule.

L3: calculates the normalized fitness.

L4: calculates the output of each fuzzy rule.

L5: calculates the final output from the fuzzy system. For more information on the model structure see [11].

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Chapter 4 Model evaluation

When evaluating a model a lot of factors have to be taken into account. A model is never a true description of a system, it is created to solve a problem. The model’s ability to solve the problem in question decides whether the model is useful or not, its ”validity”. In other words, the model’s ability to describe the system is not always the most important property, as long as it is good enough for its purpose. Sometimes it is more important that the model is not too complex, that it can be executed on its intended platform, that it is not too noise sensitive or other attributes of that kind.

When analyzing the input/output behavior of a linear model, the straightforward way would be to study the Bode plots. In the nonlinear case however, we need to run simulations on the models and study the output data. The outputs can be compared and evaluated using quality measures or by visual inspection. The methods used within this project are the cost function in Section 4.1 and visual inspection described in Section 4.2.

4.1 Cost function

The, in this project, often used accuracy measure is the cost function. The cost function is borrowed from [5] and is shown below.

Cost J = ||ym− ysim|| ||ym− ¯y||

· 100 (4.1)

Here ym is the measured output, ysim is the simulated output and ¯y is the mean of the measured output. A low value means low cost and a good fit while the value 100 means that the model is nothing better than just using the mean as estimation, note that the value could get higher than 100.

The cost provides a good measure of the models ability to describe the system but it gives no information on the complexity or stability of the model. The cost function calculation can be used on all models from which a simulation result is achievable.

4.2 Comparing the essential dynamics

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Plotting the speed-force plot with these two signals together gives a pretty good overview. A visual inspection can tell if the model is able to capture the essential dynamics of the damper without any unwanted behavior or large discrepancies. Examples of good and bad behaving models can be seen in figures 4.1 and 4.2.

It turns out to be difficult to get a model to cover the entire working area of a CES damper, even within a limited working area. Substantial knowledge of the damper dy-namics and its effects on the motorcycles road handling is therefore required in order to be able to correctly evaluate a model based on a visual inspection of the speed-force output. This project will not go into depth on the relation between the speed-force curve and the riding dynamics. The visual inspections will rather focus on more trivial matters, like the model’s ability to capture the size of the hysteresis bubble and trace the output around the zero crossing.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

Figure 4.1: A well behaving model

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

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Chapter 5 Results

The objective of this thesis project is to investigate the possibilities of creating a model to be used as an observer in a control system. This means that the model has to be able to estimate the damper’s behavior during all possible situations without losing stability. The results show that it is hard to get the models to cover the whole operating area of the damper. Really good models have been achieved for limited areas of operation.

5.1 Experiment design

A lot of different experiments have been carried out within this project. For the dis-placement part of the experiment signals containing pure sinusoids, mixtures of sinusoids, frequency sweeps and low pass filtered white noise have been tested, just to mention some of them. To these displacement signals a variety of different current signals have been tried, ranging from constant values to ramps and filtered white noise. The one mostly used however is the one generated from ¨Ohlins Racings current control solution as men-tioned in Section 3.3, from now on called reg1. The experiment design started out simple but has then been more focused on realistic operating conditions why the process has led more towards road like displacements and reg1 -kind of current inputs. The reason for this is that simple single frequency experiments have shown to be easy to model and give good model performance, but does not really contribute to the goal of achieving a good observer because of their inability to describe situations they were not modeled for. The reg1 current signal is based on a controller that operates only in a small segment of the allowed current range, 0.29 - 0.7 A. Even though most of the experiment data is based on this small segment of the current range it is still hard to get a model to cover a wide range of displacement frequencies.

5.2 Model structure

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5.2.1 Polynomial NARX model structure

The Polynomial NARX model has been the main focus in this project and has been devoted the most time. The inputs used for modeling and simulation are:

u1: The CES valve current

u2: The absolute value of the piston rod speed (the derivative of the piston rod position) u3: The sign of the piston rod speed, 1 or -1

and the output is:

y: The predicted damping force

The reason for splitting the piston rod speed from its sign is to make it possible to use non-integer powers of the speed signal in the regressors. A small collection of the resulting models are shown with their regressors below. Together with some additional models they are presented in Table 5.1 with references to figures showing their performance. The complexity field in the table indicates the number of regressors used in the model. In the figures, blue color represents the measured damping force from the dyno rig during the experiment and red color represents the model output generated with the same input signals.

Experiment Cost Complexity Figure

Sine 4 Hz 2.34 7 5.1(a)

Sine 12 Hz 1.77 10 5.1(b)

Sine 16 Hz 1.63 11 Not shown

LP filtered white noise 14.79 9 5.2

Sweep 2 - 16 Hz 6.92 5 Not shown

Sine 4 Hz on sweep model 2 - 16 Hz 63.79 5 5.3(a)

Sine 4 Hz on white noise model 136.24 9 5.3(b)

Table 5.1: Results for Polynomial NARX models

Single sine 4 Hz model

This model was generated using measurement data from a single frequency experiment. The displacement signal is a 4 Hz sine wave with 25 mm peak to peak amplitude. In this particular case the sign input u3 has been weighted and can attain the values 1 or −3.5. This weight does not improve the model in any way but is used in this example. The current signal is generated by the reg1 function. Regression vector:

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Together with its parameter vector this regression vector gives a model that performs a cost value of 2.34 when simulated for a similar validation data set. As seen in Figure 5.1(a) this model is not perfect but it behaves well and follows the measurement data with acceptable precision. Some of the curvy parts of the measurement data are here approximated as straight lines by the model and an unfortunate twitch can be noticed around the zero-crossing. In Figure 5.1(b) a similar model is shown, but created and validated on a 12 Hz sine wave. This model behaves even better and achieves a lower cost than the 4 Hz model.

Frequency sweep model

This model was generated using measurement data from a frequency sweep experiment. The displacement signal was created by sweeping a frequency from 2 to 16 Hz over a period of 30 seconds. The current signal is generated by the reg1 function. Regression vector: Reg = u1(t − 1) · u2(t − 0) · u3(t − 0) u1(t − 1)1 · u2(t − 1)1.2 u2(t − 3)0.8 y(t − 1) · u1(t − 1) · u2(t − 2)1.2· u3(t − 3) y(t − 2) · u2(t − 1)0.4· u3(t − 2)

Together with its parameter vector this regression vector gives a model that performs a cost value of 6.92 when simulated for a similar validation data set. This model validated on a 4 Hz single sine signal can be seen in Figure 5.3(a). It is quite obvious that this model is unusable for that input. The model only captures the gradients but with big offsets and almost no hysteresis.

LP filtered white noise model

This model was generated using measurement data from a white noise experiment. The displacement signal was created by low pass filtering a white noise signal with a 10th order Butterworth filter with a cutoff frequency of 15 Hz. The maximum peak to peak amplitude of the displacement was 50 mm. The current signal is generated by the reg1 function. Regression vector:

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Together with its parameter vector this regression vector gives a model that performs a cost value of 14.79 when simulated for a similar validation data set. As seen in Figure 5.2, the performance is poor. It is hard to get any valuable information out of the speed-force plot because of the cluttering. In the time domain plot it is shown that the model captures the gradients well but it does not follow the measurement data out to the extreme values. In Figure 5.3(b) it is shown that this model performs poorly when subject to an unanticipated input and does not even capture the gradients correctly.

5.2.2 Sigmoid network NARX model structure

The inputs used for modeling and simulation of the Sigmoid network NARX models are: u1: The CES valve current

u2: The piston rod speed (the derivative of the piston rod position) and the output is:

The Sigmoid network NARX models are defined by their regressors, the number of neurons in the single layer sigmoid network and an estimated parameter vector. The regression vector for these models contain all regressors allowed within the rules defined by Y , C and V . These constants determine the maximum amount of old samples of each signal, Y for y, C for u1, V for u2. For example a C value of 2 allows the regression vector to contain both u1(t − 0) and u1(t − 1). Some of the results are presented in Table 5.2 with references to figures. The data sets used are the same as in Section 5.2.1 with reg1 current input. As seen in Figure 5.5(b) and 5.6(b) the Sigmoid network NARX models behave badly when the validation data is different from the estimation data. In the figures, blue color represents the measured damping force from the dyno rig during the experiment and red color represents the model output generated with the same input signals.

Experiment Cost Y C V Neurons Figure

Sine 4 Hz 1.24 2 2 2 15 5.4

Sweep 2 - 16 Hz 2.72 2 2 2 20 5.5(a)

Sine 4 Hz on sweep model 2 - 16 Hz 22.02 2 2 2 20 5.5(b)

LP filtered white noise 11.24 2 3 3 16 Not shown

LP filtered white noise 9.86 2 4 5 30 5.6(a)

Sine 4 Hz on white noise model 38.87 2 4 5 30 5.6(b)

Table 5.2: Results for Sigmoid network NARX models

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5.2.3 Neuro-Fuzzy model structure

The inputs used for modeling and simulation of the Neuro-Fuzzy models are: u1: The CES valve current

u2: The piston rod speed (the derivative of the piston rod position) and the output is:

The Neuro-Fuzzy, or ANFIS models are here defined by the number of membership functions (MF) for each input, and the MF type. These settings are entered into the ANFIS editor when the models are created. The number of membership functions for input u1 and u2 are defined by A and B. The type of MF is chosen from a set of alternatives available in the ANFIS editor, for more information see The Fuzzy Logic toolbox for MATLAB documentation. The results are shown in Table 5.3. In the figures, blue color represents the measured damping force from the dyno rig during the experiment and red color represents the model output generated with the same input signals.

Experiment Cost A B MF Type Figure

Sine 4 Hz 2.12 4 4 gbell 5.7(a)

Sine 16 Hz 3.38 4 4 gbell 5.7(b)

Sweep 2 - 16 Hz 6.6 3 3 gbell Not shown

Sweep 2 - 16 Hz 5.2 6 6 gbell Not shown

Sine 16 Hz on Sine 4 Hz model - 4 4 gbell 5.8

Table 5.3: Results for Neuro-Fuzzy models

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(a) Model created and validated on a single 4 Hz sine wave with reg1 current

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(b) Model created and validated on a single 12 Hz sine wave with reg1 current, the green line is the reg1 reference

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-0.25-2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -1.5 -1 -0.5 0 0.5

Damping

force

[kN]

(a) Model created and validated for a LP filtered white noise signal with reg1 current

4800 4900 5000 5100 5200 5300 5400 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Time [samples] Damping force [kN]

(b) Same as above, a section of the data is presented in the time domain

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(a) Model created with a frequency sweep input and simulated with a single 4 Hz sine input, both using reg1 current

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(b) Model created with a LP filtered white noise input and simulated with a single 4 Hz sine input, both using reg1 current

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

1400 1450 1500 1550 1600 1650 1700 -2.5 -2 -1.5 -1 -0.5 0 0.5 Time [samples] Damping force [kN]

(b) Same as above, a section of the signal is presented in the time domain

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(a) Model created and validated on a 2 - 16 Hz frequency sweep with reg1 current

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(b) Same model as above but validated on a 4 Hz single frequency data set

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-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Damping

force

[kN]

(a) Model created and validated for a LP filtered white noise signal with reg1 current

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

(b) Same model as above but validated on a 4 Hz single frequency data set

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-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Damping

force

[kN]

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000

Damping

force

[N]

(b) Model created and validated on a single 16 Hz sine wave with reg1 current

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-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000

Damping

force

[N]

(a) Model created with a 4 Hz sine signal and validated with a 16 Hz sine, both with reg1 current

1100 1200 1300 1400 1500 1600 1700 -1 -0.5 0 0.5 1 1.5 x 104 Time [samples] Damping force [kN]

(b) Same as above, a section of the signal is presented in the time domain

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Chapter 6 Conclusions and future work

This thesis project comprise a comparison between different model structures and exper-iments. It does not deliver a solution that provides a stable and comprehensive model to be used as an observer in a control system but rather some ground work for further inves-tigation. An important conclusion is that regardless of used model structure, the models have limited areas of validity. This fact puts focus on the experiment design, which has to be improved in order to supply the modeling process with adequate measurement data.

6.1 Experiment design

As mentioned in Section 3.3 it is important that the experiments in some sense emulates the intended working conditions for the model. This is why the experiments presented in this report are constructed using the reg1 controller to control the current signal. The reg1 controller delivers a signal that follows the displacement signal acting a lot like what is desired from a future model based control solution. The drawback however is that the controller provided by ¨Ohlins Racing used to create the reg1 signal does not cover the entire working area but it is a reasonable limitation in the experiment design in this project. Experiments have been carried out with the current signal being a ramp through the entire allowed range but it has not provided any usable results and is far from any realistic operating situation.

It was discovered that most single frequency experiments were captured well by the models. The necessary displacement frequency range is however not that small. Model-ing with multi frequency displacement data will in some cases give usable models when validated on similar data but they are not able to perform well when the validation data is altered, even if it is altered within the frequency range for which the model was esti-mated. This leads to the question on how to design experiments that will make sure that all the features of the damper dynamics really gets planted into the model structure and parameters.

6.2 Model structure

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than the estimation data, called cross-validation. It might have potential if investigated further but that will not be recommended by this thesis.

The Polynomial NARX and the Sigmoid network NARX are more on the same level when it comes to performance. However, the Sigmoid network models achieve lower cost values for both single and multi frequency experiments and even for cross-validation. For the frequency sweep it reaches a cost value of 2.72. A value in that region would be desirable for the LP filtered white noise experiment as well. The best result right now is 9.86 and that model delivers predictions with an error magnitude of 50-100 N. The cross-validation results are lower for the Sigmoid than for the Polynomial model but still far from acceptable.

In favor of the Polynomial model structure is that the implementation of it is straight-forward and the complexity is easy to oversee. The Sigmoid model structure appears to be a little more difficult to work with. As seen in Table 5.2 for the white noise experiments, a significant increase in model complexity does not necessarily lead to a significant change in the cost value. In many cases increased complexity led to increased cost. In the Poly-nomial case an interesting phenomena was noticed for the single frequency experiments. The higher the frequency, the higher achievable complexity for the regressor selection algorithm. In opposite to the Sigmoid models, increased complexity almost always led to decreased cost values for the Polynomial models.

The performance for multi frequency experiments of course has to be improved but the biggest problem is the inability to handle cross-validation. In order to provide a stable observer for a critical control system the model must be able to cope with unexpected inputs. For the Polynomial model case, increased complexity leads to lower cost values but it also leads to a kind of overfitting. If a model gets too complex and too tied to its measurement data it will not be able to handle cross-validation. This might lead to an unsolvable equation. On one hand you want the model to be complex and accurate but on the other hand you need it to be simple in order to handle unexpected inputs and to be possible to implement in the control system. The solution is to do better experiment design. If the measurement data were to be in some sense ideal and would describe the entire operating range of the damper perfectly then the model could be allowed to gain great complexity, assuming that enough processing power is available at implementation.

6.3 Future work

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¨

Ohlins Racing has the physical knowledge of how the damper works and combining that with black-box theory might not be a bad idea. Another way of including the in-house knowledge of the dampers is to create a weighted cost function that focuses on the dy-namics making a difference. A cost function could for example give extra punishment for errors around the zero-crossing or at the end positions depending on what is considered most important for the damping performance.

Another theory that would have been interesting to try is Iterative Learning Control, ILC. This was studied within this project and considered interesting but impossible to try because of the software used to operate the dyno rig. In ILC the same experiment is iterated over and over while the model parameters are optimized for that specific ex-periment and updated in real time until sufficient performance is achieved. This requires direct interaction with the dyno rig from the modeling software which was considered too big of an operation within this project.

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Black-box modeling of a semi-active motorcycle damper