Black-box modeling of a semi-active motorcycle damper472536/FULLTEXT01.pdf · model structures to be able to capture the behavior. The development of an iterative modeling software

Black-box modeling of a semi-activemotorcycle damper

JOHN SÖDERBERG

Master’s Degree ProjectStockholm, Sweden Januari 2011

XR-EE-RT 2011:001

Abstract

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

Suspension system manufacturer Öhlins Racing AB makes a semi-active system calledCES, Continuously Controlled Electronic Suspension where the damping characteristicsare controlled by a patented electronic CES valve. This master thesis project was initiatedto create a black-box model of the CES system to be used for future control designimprovement.

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

The results show that the complex and nonlinear behavior of the CES system makesmodeling difficult, even in small areas of the operating range. The conclusions determinethat the experiments have to be limited to only the really important dynamics for themodel structures to be able to capture the behavior. The development of an iterativemodeling software connected to a laboratory test rig is also proposed.

Sammanfattning

Olika körsituationer kräver olika stötdämparinställningar, utvecklingen av stötdämpareleder därför ofta till en kompromiss mellan komfort, köregenskaper och vägh̊allning.Denna kompromiss, samt de ökande kraven p̊a mer avancerade stötdämparfunktionerhar 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 reglerasav en patenterad elektronisk CES-ventil. Detta examensarbete grundades i en önskan attskapa en black box-modell av en CES-stötdämpare för att i framtiden kunna förbättraregleringen av denna.

Det viktigaste när man skapar black box-modeller är designen av experimentet ochvalet av modellstrukturen. Detta examensarbete föresl̊ar en experimentdesign skapadför att efterlikna de önskade arbetssituationerna för CES-stötdämparen och därigenomgenerera mätdata som är relevant för en modell som ska användas till reglering. Mod-ellering sker sedan med tre olika modellstrukturer: Polynom-NARX, Sigmoid-NARX ochNeuro-Fuzzy-struktur.

Resultaten visar att CES-stötdämparens olinjära och komplexa beteende gör mod-elleringen sv̊ar, även i kraftigt avgränsade delar av dess arbetsomr̊ade. Man kan draslutsatsen att mer energi bör läggas p̊a designen av experimenten för att se till att endastrelevant mätdata n̊ar modelleringsprocessen. Examensarbetet föresl̊ar även att ett itera-tivt modelleringsverktyg kopplat direkt till en mätrigg utvecklas.

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Past work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The CES damper 42.1 Semi-active suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 A hydraulic damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 The CES valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 The control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 System identification 83.1 Black-box models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Lab equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 NARX modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4.1 Polynomial NARX models . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Sigmoid network NARX models . . . . . . . . . . . . . . . . . . . . 113.4.3 Estimating the parameters . . . . . . . . . . . . . . . . . . . . . . . 113.4.4 Automated regressor selection . . . . . . . . . . . . . . . . . . . . . 11

3.5 Neuro-Fuzzy modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Model evaluation 154.1 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Comparing the essential dynamics . . . . . . . . . . . . . . . . . . . . . . . 15

5 Results 175.1 Experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.1 Polynomial NARX model structure . . . . . . . . . . . . . . . . . . 185.2.2 Sigmoid network NARX model structure . . . . . . . . . . . . . . . 205.2.3 Neuro-Fuzzy model structure . . . . . . . . . . . . . . . . . . . . . 21

6 Conclusions and future work 306.1 Experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Bibliography 33

Chapter 1

Introduction

A conventional hydraulic damper for motor vehicle suspension needs to be designed towork under a lot of different conditions. Different driving situations demands differentdamping characteristics. Fast cornering and good handling needs a hard and stiff damp-ing while a soft damping is needed for a smooth and comfortable ride. In conventionalsuspension systems this leads to a compromise between these conflicting conditions. Asports car usually handles well, corner with minimum roll and stop with minimum brakedive allowing it to keep its traction but the hard ride makes it uncomfortable for longerdrives and bad roads. A luxury car however is soft in order to offer a smooth ride butends up suffering from bad handling. To solve this problem the vehicle industry has beendeveloping different kinds of active suspension systems over the years. The idea is thatthe damping characteristics should be continuously adjustable to better suit the currentsituation. A fully active damper can both add and remove energy from the system andtherefore 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 popularsolution is therefore to use a semi-active damper in which energy only can be absorbed, inan energy efficient way. This is usually a traditional hydraulic damper with some kind ofsystem 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 techniquesin order to get a better understanding of its behavior and enhance the performance of itscontrol system.

1.1 Background

Öhlins Racing, world famous manufacturer of advanced suspension systems, makes asemi-active suspension system called CES, Continuously Controlled Electronic Suspen-sion. This system has a hydraulic damper equipped with one or two CES valves thatelectronically controls the pressure drop between the high- and low pressure side of thedamper adjusting the damping force. Generally, the damping force delivered from ahydraulic damper depends on the speed of the piston rod, the higher the speed of thecompression or rebound movement the higher the damping force. In a CES system thedamping force of course also depends on the control current applied on the CES valve.The aim of the CES system is to, given a measured piston rod velocity, deliver the desireddamping force needed for the current situation, see Chapter 2 for more details. The CESdamper however creates a strongly nonlinear system which is very complicated to control.

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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 carmanufacturers like Audi, Volvo, Ford, Volkswagen and Mercedes-Benz. But since ÖhlinsRacings main focus is on motorcycle suspension systems they are developing the CESsystem for motorcycles as well. On a motorcycle however, the suspension system canbe made more sophisticated than most automotive systems. This is mainly because thedynamics of a motorcycle are more complicated. Another reason is that the ratio betweenthe sprung and unsprung mass is a lot smaller than on a car making it harder to obtainhigh riding comfort [1]. This makes the performance of the CES control system even moresignificant.

The damping force can easily be measured in a laboratory test rig but when thedamper is in operation on a motorcycle it becomes difficult and expensive. Since most ofthe theory regarding automatic control assume available feedback of the output signal, thismakes the control problem even more intricate. A possible way of obtaining this feedbackis to use a mathematical model of the damper acting as an observer that calculates anestimate of the damping force. Öhlins Racing are therefore interested in modeling theCES 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 identifyinga 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 findthe one that best suits the CES damper. The effect of different kinds of input data tothese 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 modelwas discovered to be too complex for execution in a real time control system. Because ofthis, 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 possibleto successfully create a black box model (or a collection of sub models) useful for controlpurposes.

Not very much research has been done on this kind of semi-active damper using valveslike the CES damper. There is however a more popular kind of damper called the MagnetoRheological (MR) damper which has been subject to quite a lot of research lately. TheMR damper has a magneto rheological fluid that changes its viscosity when an externalmagnetic field is applied. The behavior of the MR damper is in many ways similar to theCES damper and a lot of inspiration can be found in this research. In [3] a polynomialNARX model is created which seems to capture the behavior of the MR damper duringsome operational conditions. Another paper [9] compares a couple of semi-physical greybox models to a single layer neural network NARX model and finds the black box NARXmodel to be superior. In [8] different kinds of experiments are made in order to find

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out which gives the best input data for modeling of MR dampers. Common for mostof the research studied is that a good model is often achieved within some limited areaof operation. The models show good performance when simulated with clean signalssimilar to (or sometimes even the same as) the ones used for identification. Unfortunatelyno information regarding a working model based control system has been found, onlylaboratory installations and simulations.

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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 typicalspeed-force plot is shown in Figure 2.2. The speed axis represents the speed of the pistonrod, i.e. the speed with which the damper is compressed or retracted. As seen in thiscase, the higher the speed is the higher the damping force. As mentioned in Chapter1, the result is often a compromise between comfort and handling. The need to achieveboth has led to the development of active and semi-active suspension systems. The idea isthat the damping characteristics should be continuously adjustable while driving so thatthe vehicle always performs optimally. A basic system could perhaps detect the currentdriving conditions and switch between different pre-programmed speed-force curves whilea more advanced system could for example be able to detect individual bumps in the roadand momentarily reduce the damping force.

2.2 A hydraulic damper

The CES damper is a hydraulic damper equipped with an electrohydraulic valve. Theworking principle of the Öhlins prototype used in this project can be seen in Figure 2.3.During the compression stroke the oil flows from the compression chamber (A) throughthe open piston valve (B) in to the rebound chamber (C). As the piston rod enters the

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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]

Dam

ping

forc

e [k

N]

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 valveit 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. Theoil reservoir is divided into two parts separated by a dividing piston and the area (G) isfilled with nitrogen gas. This is done to avoid the risk of cavitation, that vapor bubblesare created around the piston when it is moving at high speeds, much like what happensaround a propeller in water.

2.3 The CES valve

The CES valve was invented and patented by Öhlins Racing. The valve controls a hy-draulic flow by means of a electric solenoid. The flow is adjusted by varying the electricalcurrent through the solenoid coil. This thesis will not go in to closer detail on the valveitself, it is only considered as a black box in this project. More details about the functionof the valve can be found in [7]. A cross section illustration of the valve can be seen inFigure 2.4.

5

B

D

A

C

H

G

F

E

Figure 2.3: A diagram of the CES damper

Figure 2.4: The CES valve

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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 thespeed-force characteristics while driving. This is done by altering the control current puton the CES valve. Since the control reference is a speed-force curve, the reference valuefor 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. Thedamping 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 anobserver in a control system to estimate the damping force, or be inverted and used asa feed forward controller. This project is focused on creating a model to be used as anobserver. 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.

Fref C G

Ĝ

+Fref

−

v i F

F̂

Figure 2.5: The proposed control strategy. Fref is the force reference function, C is the

controller, G is the controlled damper and Ĝ is the model based observer

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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 thesystem. The challenge is instead the choice of model structure and obtaining relevantdata. The process of identifying a system is in many ways iterative and demands sometrial and error. First of all you have to design an experiment that excites the dynamicsthat 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 differentconfigurations, sizes and orders are available. This is where most of the time is spent andthis is often automated in an iterative process of its own as will be shown later. Whenthe best model structure is determined an optimization procedure fits its parameters tothe data. The model then has to be tested and evaluated, if it is not sufficient a newiteration 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 howevergood to have some knowledge of the behavior of the system when choosing what kind ofblack-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 hashighly nonlinear dynamics, linear model structures are not sufficient as shown by earlierwork [7]. Hence only nonlinear models will be considered. The ones used are PolynomialNARX, Sigmoid network NARX and Neuro-Fuzzy models.

3.2 Lab equipment

The experiments carried out within this project were done in the laboratory at ÖhlinsRacing. The laboratory is equipped with machines to test and evaluate damper perfor-mance, mainly through so called dyno rigs. The dyno rig is basically a hydraulic actuatorwhose position is controlled by a computer through a feedback loop. A solid frame ismounted above the actuator that holds the damper’s upper mount together with a dy-namometer. The lower mount is fitted to the actuator rod, see Figure 3.1. The dyno rigis controlled and monitored by software developed at Öhlins Racing. This program can

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

Hydraulic actuator

Damper

Figure 3.1: The principle of the dynamometer rig

3.3 Experiment design

When doing system identification, the resulting model is never better than the data it isbased on. Hence obtaining good data is crucial for the whole identification process. Ofcourse the data has to be as noise free as possible, have sufficient resolution and samplingfrequency but of most importance is the experiment that the data is generated from. InLjung [4] it is proposed that the questions of where, what and when to measure is to beanswered. The question of where and what to measure is not always trivial, a systemcan have multiple inputs, outputs and disturbance signals with varying relevance. In thisparticular case however it is quite clear that the inputs should be based on the pistonrod position and valve current and that the output is the measured damping force. Thereal difficulty here lies in the question of when to measure. As mentioned earlier the

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inputs can be manipulated quite freely so the when issue is rather a question of how todesign the inputs. The experiment has to excite the dynamics that the model shouldcapture, and deciding on what is relevant for the application in question is not alwaysstraightforward. In this case, when the goal is a simulation model, one has to account forthe 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 electricalsignal is proposed. In [9] experiments are carried out with displacement signals createdfrom low pass filtering white noise, with both constant and varying electrical current. Thelow pass filtered white noise can be seen as an attempt to create a road like signal. Thisgoes hand in hand with a theory from [2] that data from the desired operating conditionsof the model is very useful for the identification process. It is also mentioned in [2] thatclosed-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 thisproject are done on a system with Öhlins Racings present control solution connected.The idea is that the main focus should be on designing the displacement input and letthe present control solution provide an electric current input signal. With this setup theexperiment data, especially the correlation between the two inputs, becomes more like inthe 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 surethat the modeling process does not have to suffer from bad data quality. The data also hasto be split into sections for estimation and validation and formatted to fit the modelingsoftware.

3.4 NARX modeling

The inputs to these model structures are called regressors. A regressor is a variablecontaining old inputs or outputs from the system, for example y(t − 1) is the system’soutput delayed one sample and u2(t − 3) is the system’s second input delayed by threesamples. The choice of regressors, called the regression vector, is really important forthe outcome of the modeling process and demands both experience and knowledge of thesystem. NARX (Nonlinear AutoRegressive model with eXogenous inputs) is a class ofnonlinear models that in some sense origin from the well known ARX models common insystem identification. As with ARX models, NARX models are discrete time input-outputequations where the output y at time k depends on old values of itself and its input u upto 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 thisproject f can be a polynomial or a sigmoid network and u represents multiple inputsu1, u2, . . . , um.

Regardless of the model structure used it is always recommended to have differentdata sets for estimation and validation of the model. If the same data set is used both toestimate and validate the model parameters there is a risk of overfitting [6]. This meansthat the model is getting adjusted to the specific noise realization apparent in the data

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set. This overfit is of no benefit to the model since the model will be subject to differentnoise 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 structureis called a polynomial NARX model. The nonlinear regressors of the polynomial NARXmodel are products of powers of the earlier mentioned regressors. A nonlinear regressorcould 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 twoinput signals to the system. The full polynomial NARX model will be on the form:

y(t) = a0 +N∑i=1

ai · Regi (3.3)

Where ai is a weight. This kind of model structure makes computer implementationstraightforward which makes them well suited for model-based control. In [3] it is shownthat polynomial NARX models can exhibit a wide variety of hysteretic behaviors similarto 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 additionalfabricated 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 networkhas a structure like:

g(x) =n∑

k=1

αkK(βk(x− γk)) (3.4)

Where K(s) = (es+1)−1 is the sigmoid function, βk is 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 isdecided by n. For more information see [5]. It is possible to use the nonlinear regressorsfrom Section 3.4.1 as inputs to this model structure as well but it has not shown to be ofany benefit in this project.

3.4.3 Estimating the parameters

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

3.4.4 Automated regressor selection

As mentioned earlier, selecting the best set of regressors is a difficult task. Several methodsof selecting the regressor set automatically have been proposed but none of them guar-antees to give the optimal solution. The iterative method described here is mentioned in[12], a flow diagram is shown in Figure 3.2.

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Initialize- Measurement data- Candidate regressors- Initial model

Evaluate model- Calculate cost function

Add random regressorto the model and remove it

from the candidate regressor set

Final model

Stop?No

Yes

Evaluate model- Calculate cost function

Better?

Remove latest regressorfrom 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

Figure 3.2: Flow diagram of the automatic regressor selection

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Initialization

The algorithm has to be initialized, this is done by supplying the measurement data inan appropriate format, generating the set of candidate regressors and picking the initialmodel. The measurement data has to be in a format compatible with the software that isgoing to be used for parameter estimation and simulation. The set of candidate regressorsis then generated by creating a list of the allowed outputs/inputs with their respectivedelays. This will do for the sigmoid NARX model structure but when this method isapplied on the polynomial NARX models the candidate regressor set is a bit different. Inthat case the list of candidate regressors is used to create a new list of nonlinear regressorsby combining the regressors with different delays and powers according to predeterminedrules. These rules have to be restrictive as the amount of nonlinear regressors easily cangrow out of proportion and make the process unnecessarily time consuming. Regardlessof 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 dataand the model is then simulated using the validation data. The result of the simulationis 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. Thecost function is borrowed from [5] and is shown below.

Cost J =||ym − ysim||||ym − ȳ||

· 100 (3.5)

Here ym is the measured output, ysim is the simulated output and ȳ is the mean of themeasured output. A low value means low cost and a good fit while the value 100 meansthat the model is nothing better than just using the mean as estimation, note that thevalue could get higher than 100. The cost value makes it straightforward to decide if anew 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 removedfrom that list to avoid reusing it. The regressor is then added to the regression vectorand the resulting model is evaluated. If the new regressor leads to a reduced cost it isaccepted and the pruning begins, if not it is removed and a new regressor is picked.

Pruning

In the pruning step the algorithm tries to reduce the amount of regressors in the regressionvector. This is not only done to keep the size of the vector reasonable but also becauseadding new regressors sometimes make older ones redundant, especially in the polynomialNARX models. Pruning often leads to both reduced cost and regression vector size. Inthis process one regressor at a time is removed from the regression vector and the resultsare evaluated. If one of the reduced models yields a lower cost it becomes the new modeland the pruning starts over. This is repeated until none of the reduced models offers alower cost.

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Stop criterion

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

3.5 Neuro-Fuzzy modeling

Within this project some studies were made on neuro-fuzzy models. The Fuzzy Logictoolbox for MATLAB includes tools for working with ANFIS (Adaptive Neuro-FuzzyInference Systems). This is a kind of model structure that combines fuzzy logic withneural networks and is said to be able to emulate most nonlinear systems. The ANFIStool uses a learning algorithm based on back propagation gradient descent and leastsquares methods to iteratively create a fuzzy inference system [10]. This thesis will notgo into depth in the fuzzy logic theory, the interested reader is referred to literature onthe subject. The ANFIS tool has an easy to use interface where the model structureparameters are tuned. The modeling procedure is mostly trial and error but the choice ofmodel size is obviously a tradeoff between precision and computational cost. The ANFISstructure can be seen in Figure 3.3.

All the parameters in (1) are shown as follows

f0 is the o�set force of the MR damper, cb is the slope coe�cient of the hysteresis curve, fy and k are two coe�cients characterizing the maximal damping force, and cw is the width coe�cient of the hysteresis curve, respectively. Fig. 1 shows the relation for the input voltages 0V, 2V, 4V and 7.5V, respectively when f0 =20, cb =1, fy =300, k=1 and cw =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 �rst-order TSK fuzzy system and ANFIS structure If a �rst-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 A1 and y is B1, then z1=p1x+q1y+r 1Rule 2: If x is A2 and y is B2, then z2=p2x+q2y+r 2

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

Where x,y are input language variables. A1,A2,B 1 and B2 are fuzzy sets. z1,z2 are output language variables. p1,q1,r 1,p2,q2and r2 are the output parameters of the fuzzy system.

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

1 2,W W mean the �tness of the fuzzy rule, 1 2,W W denote the normalized �tness.

Figure 3. ANFIS structure of �rst-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 �tness of each fuzzy rule. L3: solving the normalized �tness. 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 �rst-order TSK fuzzy model consists of three inputs and one output, the ist IF-THEN fuzzy rule reads

Rule i If u is Ai and u is Bi and V is Cithen, yi=pi u +qi u +ti V +r i

where Ai,B i and Ci are fuzzy sets, u , u and V are input language variables, yi is output language variables, pi,qi,tiand ri are the output parameters of fuzzy system.

V

u

u

fMR

Damper

ANFIS of direct model

+

−

•

•

f̂•

W 2

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z1=p1x+q 1y+r

z2=p2x+q 2y+r

1 1 2 21 1 2 2

1 2

W z W zz W z W z

W W+

= = ++

x y

X

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1 .1( 2.3 )1y

yl V

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+

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+

1.041 10.34b

b V

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+

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z

x y

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A1

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L2 L3 L4 L5

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z1

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461465

Authorized licensed use limited to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on August 16,2010 at 12:20:27 UTC from IEEE Xplore. Restrictions apply.

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. Thestructure 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 nevera true description of a system, it is created to solve a problem. The model’s ability tosolve the problem in question decides whether the model is useful or not, its ”validity”. Inother words, the model’s ability to describe the system is not always the most importantproperty, as long as it is good enough for its purpose. Sometimes it is more importantthat the model is not too complex, that it can be executed on its intended platform, thatit is not too noise sensitive or other attributes of that kind.

When analyzing the input/output behavior of a linear model, the straightforwardway would be to study the Bode plots. In the nonlinear case however, we need to runsimulations on the models and study the output data. The outputs can be compared andevaluated using quality measures or by visual inspection. The methods used within thisproject 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 functionis borrowed from [5] and is shown below.

Cost J =||ym − ysim||||ym − ȳ||

· 100 (4.1)

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

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

4.2 Comparing the essential dynamics

As mentioned in Section 2.1 the performance of a hydraulic damper can be defined by itsspeed-force plot. When comparing the model output to the actual measurement outputit is preferably done in the speed-force domain rather than in the time-force domain.

15

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 thedamper without any unwanted behavior or large discrepancies. Examples of good andbad 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 CESdamper, 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 beable 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 theriding dynamics. The visual inspections will rather focus on more trivial matters, like themodel’s ability to capture the size of the hysteresis bubble and trace the output aroundthe zero crossing.

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Figure 4.1: A well behaving model

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Figure 4.2: A badly behaving model

16

Chapter 5

Results

The objective of this thesis project is to investigate the possibilities of creating a modelto be used as an observer in a control system. This means that the model has to be ableto 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 ofthe damper. Really good models have been achieved for limited areas of operation.


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 someof them. To these displacement signals a variety of different current signals have beentried, ranging from constant values to ramps and filtered white noise. The one mostlyused however is the one generated from Öhlins Racings current control solution as men-tioned in Section 3.3, from now on called reg1. The experiment design started out simplebut has then been more focused on realistic operating conditions why the process has ledmore towards road like displacements and reg1 -kind of current inputs. The reason forthis is that simple single frequency experiments have shown to be easy to model and givegood model performance, but does not really contribute to the goal of achieving a goodobserver because of their inability to describe situations they were not modeled for. Thereg1 current signal is based on a controller that operates only in a small segment of theallowed current range, 0.29 - 0.7 A. Even though most of the experiment data is basedon this small segment of the current range it is still hard to get a model to cover a widerange of displacement frequencies.

5.2 Model structure

The model structures that were compared show similar properties. They all work prettywell for limited frequency displacement signals and show increased cost values as soon asthe input spectrum increases.

17

5.2.1 Polynomial NARX model structure

The Polynomial NARX model has been the main focus in this project and has beendevoted 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 usenon-integer powers of the speed signal in the regressors. A small collection of the resultingmodels are shown with their regressors below. Together with some additional modelsthey are presented in Table 5.1 with references to figures showing their performance. Thecomplexity field in the table indicates the number of regressors used in the model. Inthe figures, blue color represents the measured damping force from the dyno rig duringthe experiment and red color represents the model output generated with the same inputsignals.

Experiment Cost Complexity FigureSine 4 Hz 2.34 7 5.1(a)Sine 12 Hz 1.77 10 5.1(b)Sine 16 Hz 1.63 11 Not shownLP filtered white noise 14.79 9 5.2Sweep 2 - 16 Hz 6.92 5 Not shownSine 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 thisparticular 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. Thecurrent signal is generated by the reg1 function. Regression vector:

Reg = u2(t− 0) · u3(t− 0)u1(t− 1) · u2(t− 3)0.4 · u3(t− 3)u2(t− 1)1.2 · u3(t− 1)u1(t− 1) · u2(t− 3)y(t− 1) · u1(t− 1) · u2(t− 1)0.2 · u3(t− 2)y(t− 1) · u1(t− 0) · u2(t− 0)0.2 · u3(t− 2)u1(t− 1) · u2(t− 0)0.8

18

Together with its parameter vector this regression vector gives a model that performs acost value of 2.34 when simulated for a similar validation data set. As seen in Figure5.1(a) this model is not perfect but it behaves well and follows the measurement datawith acceptable precision. Some of the curvy parts of the measurement data are hereapproximated as straight lines by the model and an unfortunate twitch can be noticedaround the zero-crossing. In Figure 5.1(b) a similar model is shown, but created andvalidated on a 12 Hz sine wave. This model behaves even better and achieves a lower costthan 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 aperiod of 30 seconds. The current signal is generated by the reg1 function. Regressionvector:

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 acost value of 6.92 when simulated for a similar validation data set. This model validatedon a 4 Hz single sine signal can be seen in Figure 5.3(a). It is quite obvious that thismodel is unusable for that input. The model only captures the gradients but with bigoffsets and almost no hysteresis.

LP filtered white noise model

This model was generated using measurement data from a white noise experiment. Thedisplacement signal was created by low pass filtering a white noise signal with a 10thorder Butterworth filter with a cutoff frequency of 15 Hz. The maximum peak to peakamplitude of the displacement was 50 mm. The current signal is generated by the reg1function. Regression vector:

Reg = u1(t− 0) · u2(t− 0)0.6 · u3(t− 1)y(t− 2) · u1(t− 1) · u2(t− 0) · u3(t− 0)y(t− 2) · u1(t− 0) · u2(t− 1) · u3(t− 1)y(t− 2) · u2(t− 2) · u3(t− 0)u1(t− 0) · u2(t− 1)0.2 · u3(t− 3)u1(t− 1) · u2(t− 3)0.6 · u3(t− 1)u2(t− 3)u2(t− 1)0.6

u1(t− 1) · u2(t− 1)0.6 · u3(t− 1)

19

Together with its parameter vector this regression vector gives a model that performsa cost value of 14.79 when simulated for a similar validation data set. As seen in Figure5.2, 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 modelcaptures the gradients well but it does not follow the measurement data out to the extremevalues. In Figure 5.3(b) it is shown that this model performs poorly when subject to anunanticipated 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:


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

Experiment Cost Y C V Neurons FigureSine 4 Hz 1.24 2 2 2 15 5.4Sweep 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 shownLP 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

In Figure 5.4 it can be seen that the Sigmoid network NARX model works better thanthe Polynomial NARX model for the 4 Hz single frequency experiment. This is the lowestcost value recorded. Figure 5.5(a) shows estimation and validation for a frequency sweep.Good performance for that validation data but clearly unusable when validated for adifferent data set as seen in Figure 5.5b. Some of the LP filtered white noise experimentsare shown in Figure 5.6. The cluttered Figure 5.6(a) shows that the model captures mostof the behavior but with offsets, it is however useless for different validation data as seenin Figure 5.6(b).

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5.2.3 Neuro-Fuzzy model structure

The inputs used for modeling and simulation of the Neuro-Fuzzy models are:


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 membershipfunctions (MF) for each input, and the MF type. These settings are entered into theANFIS editor when the models are created. The number of membership functions forinput u1 and u2 are defined by A and B. The type of MF is chosen from a set ofalternatives available in the ANFIS editor, for more information see The Fuzzy Logictoolbox 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 experimentand red color represents the model output generated with the same input signals.

Experiment Cost A B MF Type FigureSine 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 shownSweep 2 - 16 Hz 5.2 6 6 gbell Not shownSine 16 Hz on Sine 4 Hz model - 4 4 gbell 5.8

Table 5.3: Results for Neuro-Fuzzy models

As can be seen in the Figure 5.7 the Neuro-Fuzzy models works well when validatedon similar data but as soon as you expose the models to something unexpected they cannot be trusted. In Figure 5.8 it is clearly shown that a change of frequency is enough tomake the model unstable, it was not even possible to calculate a cost value. Because ofthis, not a lot of effort has been put into developing these models any further.

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

Figure 5.1: Polynomial NARX model for single frequency experiments, blue color is mea-sured output and red color is model output

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Figure 5.2: Polynomial NARX models for LP filtered white noise, blue color is measuredoutput and red color is model output

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

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Figure 5.3: Results of Polynomial NARX models with inputs different from the modelinginput, blue color is measured output and red color is model output

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Figure 5.4: Results of Sigmoid network NARX models on a single 4 Hz sine wave, bluecolor is measured output and red color is model output

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

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Figure 5.5: Results of Sigmoid network NARX model created with a frequency sweepdata set, blue color is measured output and red color is model output

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Figure 5.6: Results of Sigmoid network NARX model created with a LP filtered whitenoise data set, blue color is measured output and red color is model output

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Figure 5.7: Results of Neuro-Fuzzy models on single frequency sine waves, blue color ismeasured output and red color is model output

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Figure 5.8: Results of a Neuro-Fuzzy model estimated for a 4 Hz sine wave and validatedwith a 16 Hz sine wave, blue color is measured output and red color is model output

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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 tobe 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 modelshave limited areas of validity. This fact puts focus on the experiment design, which has tobe improved in order to supply the modeling process with adequate measurement data.


As mentioned in Section 3.3 it is important that the experiments in some sense emulatesthe intended working conditions for the model. This is why the experiments presented inthis report are constructed using the reg1 controller to control the current signal. Thereg1 controller delivers a signal that follows the displacement signal acting a lot like whatis desired from a future model based control solution. The drawback however is thatthe controller provided by Öhlins Racing used to create the reg1 signal does not coverthe entire working area but it is a reasonable limitation in the experiment design in thisproject. Experiments have been carried out with the current signal being a ramp throughthe entire allowed range but it has not provided any usable results and is far from anyrealistic operating situation.

It was discovered that most single frequency experiments were captured well by themodels. The necessary displacement frequency range is however not that small. Model-ing with multi frequency displacement data will in some cases give usable models whenvalidated on similar data but they are not able to perform well when the validation datais 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 thatall the features of the damper dynamics really gets planted into the model structure andparameters.

6.2 Model structure

The three kinds of models evaluated in this project have despite their different structuresshown quite similar behavior. Unfortunately it is the bad performance that stands out,for some more than others. In the Neuro-Fuzzy case, interest was dropped in a quite earlystage because of its unacceptable behavior for validation data from a different experiment

30

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

The Polynomial NARX and the Sigmoid network NARX are more on the same levelwhen it comes to performance. However, the Sigmoid network models achieve lower costvalues for both single and multi frequency experiments and even for cross-validation. Forthe frequency sweep it reaches a cost value of 2.72. A value in that region would bedesirable for the LP filtered white noise experiment as well. The best result right nowis 9.86 and that model delivers predictions with an error magnitude of 50-100 N. Thecross-validation results are lower for the Sigmoid than for the Polynomial model but stillfar 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 bea 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 changein 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 selectionalgorithm. In opposite to the Sigmoid models, increased complexity almost always led todecreased cost values for the Polynomial models.

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

6.3 Future work

Even though it feels like everything has been tested within this project, there is alwaysmore to do. Some ideas have come up during this thesis project that were never tested.Some tests were made using more input signals. An extra input does however instantlylead to increased complexity and it is not always worth it. Using the acceleration (orrather a low pass filtered second derivative of the position) as an input was tried butdid not lead to any significant improvements. It is not recommended to use a secondderivative of a measured signal since the measurement noise can grow out of proportionbut it might be a good idea to investigate the use of other inputs. Maybe some kind ofconstructed input that provides an estimation of the displacement frequency or the sizeof the hysteretic bubble from a look-up table. Constructed inputs of this kind could be agood complement to the model structures investigated in this project but will eventuallylead to grey-box models rather than black-box models. There are a lot of papers availableon grey-box modeling of semi-active dampers and that might be another possible solution.

31

Öhlins Racing has the physical knowledge of how the damper works and combining thatwith black-box theory might not be a bad idea. Another way of including the in-houseknowledge 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 forerrors around the zero-crossing or at the end positions depending on what is consideredmost 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 totry because of the software used to operate the dyno rig. In ILC the same experimentis 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 requiresdirect interaction with the dyno rig from the modeling software which was considered toobig of an operation within this project.

A common proposition when the resulting models have a limited area of operation isto do sub models, smaller models that operate only in limited areas of the total operatingrange. This would however lead to a lot of difficulties like where to define the limits ofthe sub models and how to handle the transitions between them. Possible sub modelscould be for different areas of the valve current range, for different piston rod speeds orfor different areas of any above mentioned constructed input. It could be done but itwould demand both physical knowledge of the system and advanced stability analysis ofthe transitions.

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Bibliography

[1] G. Cocco. Motorcycle design and technology. Vimodrone (MI), 1999.

[2] H. Hjalmarsson. From experiment design to closed loop control. Automatica,41(3):393–438, March 2005.

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