Local Sensitivity Analysis of Nonlinear Models - Applied to Aircraft Vehicle Systems

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Local Sensitivity Analysis of Nonlinear

Models – Applied to Aircraft Vehicle

Systems

Ylva Jung

Division of Fluid and Mechanical Engineering Systems

Degree Project

Department of Management and Engineering

LIU-IEI-TEK-A--09/00707--SE

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Lokal känslighetsanalys av icke-linjära modeller –

tillämpat på grundflygplansystem

Examensarbete utfört i Fluid och mekanisk systemteknik

vid Tekniska högskolan i Linköping

av

Ylva Jung

LIU-IEI-TEK-A--09/00707--SE

Handledare: Sören Steinkellner

SAAB AB

Petter Krus

IEI, Linköpings universitet

Examinator: Petter Krus

IEI, Linköpings universitet

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Avdelning, Institution

Division, Department

Division of Fluid and Mechanical Engineering Sys-tems

Department of Management and Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2009-10-23 Språk Language Svenska/Swedish Engelska/English ⊠ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ⊠

URL för elektronisk version

http://www.iei.liu.se/flumes?l=en http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51212 ISBN — ISRN LIU-IEI-TEK-A--09/00707--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

Lokal känslighetsanalys av icke-linjära modeller – tillämpat på grundflygplansys-tem

Local Sensitivity Analysis of Nonlinear Models – Applied to Aircraft Vehicle Sys-tems Författare Author Ylva Jung Sammanfattning Abstract

As modeling and simulation becomes a more important part of the modeling process, the demand on a known accuracy of the results of a simulation has grown more important. Sensitivity analysis (SA) is the study of how the variation in the output of a model can be apportioned to different sources of variation. By performing SA on a system, it can be determined which input/inputs influence a certain output the most. The sensitivity measures examined in this thesis are the Effective Influence Matrix, EIM, and the Main Sensitivity Index, MSI.

To examine the sensitivity measures, two tests have been made. One on a lab-oratory equipment including a hydraulic servo, and one on the conceptual landing gear model of the Gripen aircraft. The purpose of the landing gear experiment is to examine the influence of different frictions on the unfolding of the landing gear during emergency unfolding. It is also a way to test the sensitivity analysis method on an industrial example and to evaluate the EIM and MSI methods.

The EIM and MSI have the advantage that no test data is necessary, which means the robustness of a model can be examined early in the modeling process. They are also implementable in the different stages of the modeling and simulation process. With the SA methods in this thesis, documentation can be produced at all stages of the modeling process. To be able to draw correct conclusions, it is essential that the information that is entered into the analysis at the beginning is well chosen, so some knowledge is required of the model developer in order to be able to define reasonable values to use.

Wishes from the model developers/users include: the method and model qual-ity measure should be easy to understand, easy to use and the results should be easy to understand. The time spent on executing the analysis has also to be well spent, both in the time preparing the analysis and in analyzing the results.

The sensitivity analysis examined in this thesis display a good compromise between usefulness and computational cost. It does not demand knowledge in programming, nor does it demand any deeper understanding of statistics, making it available to both the model creators, model users and simulation result users.

Nyckelord

Keywords Local sensitivity analysis, Effective Influence Matrix, Main Sensitivity Index, Sen-sitivity measures, Uncertainties, Dymola, Hopsan, Vehicle systems

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Abstract

As modeling and simulation becomes a more important part of the modeling process, the demand on a known accuracy of the results of a simulation has grown more important. Sensitivity analysis (SA) is the study of how the variation in the output of a model can be apportioned to different sources of variation. By performing SA on a system, it can be determined which input/inputs influence a certain output the most. The sensitivity measures examined in this thesis are the Effective Influence Matrix, EIM, and the Main Sensitivity Index, MSI.

To examine the sensitivity measures, two tests have been made. One on a lab-oratory equipment including a hydraulic servo, and one on the conceptual landing gear model of the Gripen aircraft. The purpose of the landing gear experiment is to examine the influence of different frictions on the unfolding of the landing gear during emergency unfolding. It is also a way to test the sensitivity analysis method on an industrial example and to evaluate the EIM and MSI methods.

The EIM and MSI have the advantage that no test data is necessary, which means the robustness of a model can be examined early in the modeling process. They are also implementable in the different stages of the modeling and simulation process. With the SA methods in this thesis, documentation can be produced at all stages of the modeling process. To be able to draw correct conclusions, it is essential that the information that is entered into the analysis at the beginning is well chosen, so some knowledge is required of the model developer in order to be able to define reasonable values to use.

Wishes from the model developers/users include: the method and model qual-ity measure should be easy to understand, easy to use and the results should be easy to understand. The time spent on executing the analysis has also to be well spent, both in the time preparing the analysis and in analyzing the results.

The sensitivity analysis examined in this thesis display a good compromise between usefulness and computational cost. It does not demand knowledge in programming, nor does it demand any deeper understanding of statistics, making it available to both the model creators, model users and simulation result users.

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Acknowledgments

This is my master’s thesis for a degree of Master of Science in Applied Physics and Electrical Engineering. It has been carried out at the section for Simulation and Thermal Analysis at Saab Aerosystems, Saab AB in Linköping, in cooperation with the department of Management and Engineering (IEI) at the University of Linköping.

Some thanks are needed. Thanks to everyone at TDGT and FluMeS/Machine design at IEI for making feel welcome. Special thanks to my supervisor Sören Steinkellner for giving me the chance to do this thesis, as well as all the help, the answers given and for always taking the time. Thanks also to my examiner Petter Krus for valuable inputs and helping me with HOPSAN and Niclas Wiker for all the help on the landing gear model and Dymola.

Last but not least Daniel, for listening to all my worries about the thesis and the future, for believing in me, making me believe in me and for making me feel loved.

Ylva Jung, Linköping October 2009

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Introduction

This first chapter gives a brief introduction to the company background and the problem background in section 1.1, as well as the goals and aims of this thesis in sections 1.2 and 1.3 respectively. Section 1.4 presents the method used and section 1.5 contains a reader’s guide.

1.1 Background

1.1.1 Company background

Saab is a Swedish high-technological company with focus on defense, aviation and civil security, and in the beginning the company was manufacturing planes. “Sven-ska Aeroplan Aktiebolaget” (Swedish for “Swedish Aeroplane Limited”) (SAAB) was founded in 1937 in Trollhättan, has had the headquarters in Linköping for many years but is now based in Stockholm. In 1965 the name was changed to Saab AB.

The light bomber and reconnaissance aircraft B17 was the company’s first aircraft, and with it Saab became the leading supplier to the Swedish Air Force. The J29 Tunnan fighter was introduced in the late 1940’s, followed by Lansen (1950’s), Draken (1960) and Viggen (1971). JAS 39 Gripen entered service in 1993. During the 1980’s investments in civil aircrafts were made resulting in Saab 340 and Saab 2000.

Besides military and commercial aircraft production, automobile manufactur-ing began in the late 1940’s and in the 1960’s Saab helped to create Sweden’s computer, missile and space industries. In 1969 Saab and Scania merged to Saab-Scania, manufacturing automobiles, trucks and buses. Since 1990 the passenger car division is an independent company, Saab Automobile, and in 1995 Saab-Scania was demerged into two companies, with bus and truck constructor Saab-Scania separated from Saab.

Saab has also, by acquisitions of the defense group Celsius in 2000, diversified into defense industry with roots in companies like Bofors, Philips, Datasaab, Eric-sson, AGA and Satt Electronics. The product-range is focused on future defense

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4 Introduction

needs and safer society [18].

This thesis is done at Saab Aerosystems in Linköping, at the section for Sim-ulation and Thermal Analysis, TDGT. This group has 15 employees and works primarily with the Gripen’s Vehicle Systems, i.e. fuel system, auxiliary power unit and system, environmental control system, hydraulic system, landing gear and rescue system. The section also works with other Saab aircrafts and aircrafts from other manufacturers as well as UAVs. The work consists of:

• analyses of vehicle systems including regulation of the hardware and soft-ware, e.g. simulations of new systems and calculations of the performance of completed systems.

• development of calculation models and real time models in vehicle systems. • CFD calculations of the internal currents and temperature fields of the

air-craft interior and the vehicle systems. • R&D within the field of vehicle systems.

Modeling and simulation within vehicle systems are used today for:

• total system specification and design, e.g. functionality on the ground and in the air.

• equipment specification and design. • software specification and design. • various simulators.

• test rig design.

1.1.2 Problem background

In order to minimize the time of development for a product as well as reducing the necessity of extended testing of the physical product, the demand on a known accuracy (see definition A.1) of the results of a model simulation (i.e. how well the result reflects the behaviour of a real system) has grown more important. Uncer-tainties in system parameters (e.g. weight, length, volume) entail uncerUncer-tainties in the result of a simulation. The accuracy of a simulation result is a function of the knowledge of the uncertainties in the system parameters, which can be estimated with or without access to test data. It is also based on the knowledge that there are other sources of uncertainties, unknown to us.

1.2 Goal

The goal of the thesis is to evaluate different measures of accuracy of the results for physical models so as to decide how useful they are in practice for the kind of models developed at TDGT.

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1.3 Aims 5

1.3 Aims

Practical needs for TDGT on a measure of accuracy are:

• to be useful in simulations of both static and dynamic systems.

• to work at the early stages of conceptual design (uncertainties are rough guesses) as well as when test data is available.

• to serve as model documentation for the uncertainties of system parameters. • to provide synergy effects e.g. possibility to point out which model

compo-nent contribute the most to the inaccuracy in simulation results. • to fit with already implemented system model components.

If the needs above are fulfilled by a measure of accuracy and the method is implementable in a company development process, the development of new systems will be easier and the quality of simulation models and results of calculation will improve. In the long run this will lead to reduced costs in developing aircrafts.

1.4 Method

This thesis started out with a litterature review to deepen the knowledge about the theories behind sensitivity analysis. The decision was made to proceed with the local sensitivity analysis methods EIM (see section 3.2.1) and MSI ( 3.2.2). To evaluate the possibility to implement these at TDGT, tests have been made; one on a physical model of a hydraulic servo and one on a model developed and used at TDGT, the conceptual model of a Gripen landing gear.

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6 Introduction

1.5 Reader’s guide

Chapter 1 Introduction to the thesis, background, goals and aims of the thesis.

Chapter 2 This chapter presents the theoretical background to model-ing and simulation. It describes the profits of modelmodel-ing and simulation, different types of modeling and problems that will emerge during the modeling process, such as uncertain-ties and disturbances. There is also a short presentation of models used at TDGT.

Chapter 3 In this chapter, the theory behind Sensitivity Analysis (SA) is presented.

Chapter 4 Two experiments have been made to test the SA method, they are presented here.

Chapter 5 Results of the hydraulic servo experiment.

Chapter 6 Results of the landing gear experiment.

Chapter 7 Summary and conclusions.

Chapter 8 Ideas and suggestions for future improvements are pre-sented here.

Bibliography

Appendix A Definitions and abbreviations used in the thesis.

Appendix B User’s guide.

Appendix C Some problems that have occurred during the work with this thesis, and their causes are listed.

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Chapter 2

Theory - Modeling and

simulation

Simulation grows more and more important in development processes today, and in order to get reliable results, sufficiently good models are needed.

This chapter introduces the theoretical background of modeling and simulation (M&S) as presented in this thesis. It starts with some definitions in section 2.1, and describes the profits of modeling and simulation in section 2.2. The physical modeling and system identification methods are introduced in section 2.3, some model properties in section 2.4 and the (M&S) tools Simulink and Dymola in section 2.5. Then the concepts of uncertainties and disturbances are described in sections 2.6 and 2.7, followed by a short description on how models are used in control engineering in section 2.8. The chapter ends with a survey of the models used at TDGT in section 2.9.

Someone with knowledge about modeling and simulation probably need not read parts of this chapter and could concentrate on section 2.4.4 about the level of model validity, section 2.6 about uncertainties and section 2.9 about the models used at TDGT.

2.1 Some definitions

Many definitions exist for the terms used in this thesis; the following definitions all come from [5] if nothing else is stated.

System:

A system is a potential source of data.

Another way of describing a system is with an example.

The largest possible system of all is the universe. Whenever we de-cide to cut out a piece of the universe such that we can clearly say

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8 Theory - Modeling and simulation

what is inside that piece (belongs to that piece), and what is outside (does not belong to that piece), we define a new system. A system is characterized by the fact that we can say what belongs to it and what does not, and by the fact that we can specify how it interacts with its environment. System definitions can furthermore be hierarchical. We can take the piece from before, cut out a yet smaller part of it, and we have a new system.

Inputs and outputs:

Closely linked to a system are inputs and outputs. Inputs are variables that are generated by the environment and that influence the behaviour of the system.

Outputsare variables that are determined by the system and that in turn influence

the behaviour of its environment.

Experiment:

An experiment is the process of extracting data from a system by exerting it through its outputs.

Model:

A model (M) for a system (S) and an experiment (E) is anything to which E can be applied in order to answer questions about S.

A model is a delimitation of the aspects of the properties of a system that are relevant for a specific aim. A model is always a simplification of the real system. Since a model is determined by the aim and the knowledge about the system, a model can never be “true” or “correct”. It only has to be good enough for the defined system/problem at hand. A model of a system can thus be valid for one experiment and invalid for another.

Simulation:

A simulation is an experiment performed on a model.

Or, as presented in [2]:

Simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of the system and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

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2.2 The benefits of modeling and simulating 9

2.2 The benefits of modeling and simulating

So what are the benefits of simulation, and where does modeling come into the picture?

There are many reasons to do a simulation. As is implied in the definition above, section 2.1, simulation is a way of gaining knowledge about a physical or abstract system without the real system. Situations where it would be expensive to use the real system, or when this could cause danger or even death are areas where simulation is used. These typically include simulation of technology for performance optimization, safety engineering, testing, training and education, e.g. in the domain of nuclear power. Another useful application of simulation is to test products that do not yet exist, i.e. in order to evaluate different designs without the need to produce expensive prototypes. Simulations can also be used to “speed up” or “slow down” time, in cases where the time lapses are too fast (e.g. an explosion) or too slow (e.g. the creation of a galaxy). In addition to time scaling there is also the possibility of size scaling.

Another advantage of simulation over experiments with the real system is that all inputs and outputs are available. Moreover the disturbances are also accessible, making it possible to determine in a more precise way the influence of different elements.

Simulation is also a possible way of gaining knowledge and detecting problems before they occur. With a simulation of a diving aircraft it is for example possible to calculate the forces on the wings, thus determining the thickness needed. This helps making correct design decisions at an early stage and consequently reduce costs, see figure 2.1.

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2.3 Modeling methods for physical systems

When starting the modeling process there are different ways to go, two types of models are physical models, also called mathematical models, and models con-structed through System Identification. Regardless of which, the process is itera-tive and the modeler has to continue until the model is sufficiently good for the purpose it was intended for.

Most models also go through the phases of verification and validation. Veri-fication is a determination of whether the computational implementation of the conceptual model is correct. Validation is the determination of whether the con-ceptual model can be substituted for the real system for the purposes of experi-mentation [2]. Or in two short questions. Verification: Did I build the thing right? Validation. Did I build the right thing?

Two of the more commonly used modeling methods for physical systems are described below.

2.3.1 Physical modeling

Physical models, or mathematical models, are based on mathematical equations describing the physical behaviour of the system being modeled. The relations be-tween the model variables and signals are expressed as mathematical connections, these are often differential equations.

One big advantage of physical modeling is that the underlying physical relations still are visible for the modeler, and the physical variables can be distinguished.

2.3.2 System Identification

System identification (SI) is a modeling method that results in a so called black box model, where only the relation between the inputs and outputs are described. With this method the model is not based on the fundamental physical relations. The description of the system modeled can be mathematical or in the form of a graph or a table. It can e.g. be a step response, an impulse response or a Bode plot.

An advantage of system identification is that the final model can be less com-plex than when using for example physical modeling.

2.3.3 Mixing modeling methods

It is possible to mix the use of different modeling methods, where parts of the model can be based on physical modeling and others on system identification. This is done to benefit from the advantages of the different methods.

Physical models are sometimes called white box models, and with SI being black box modeling, as a result there is something called grey-box model. One example is models consisting of differential equations where several parameters occurring in these equations are not known and have to be extracted from the system by data-based methods [13].

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2.4 Model properties 11

2.4 Model properties

2.4.1 Linearity

Linearity is one of the most important concepts in mathematical modeling. Models of devices or systems are said to be linear when their basic equations — whether algebraic, differential or integral — are such that the magnitude of their or response produced is directly proportional to the excitation or input that drives them [11]. A linear function f (x) is such that it satisfies both of the following properties: additivity and homogeneity. Additivity demands that f (x + y) = f (x) + f (y) and homogeneity that f (αx) = αf (x).

Or, as presented in [16] with an example.

˙x = f (x, u)

y = h(x, u) (2.1)

The model described by equation (2.1) is said to be linear if f (x, u) and h(x, u) are linear functions of the inner states, x, and the inputs, u:

f (x, u) = Ax + Bu

h(x, u) = Cx + Du (2.2)

For a linear system excited by a complex set of inputs, it is possible to use the principle of superposition. This is to say that the output of the system is the same regardless whether the system is exposed to the input as it is, or if the input is divided into smaller inputs, each exciting the system, and these are added, or superposed, see figure 2.2.

Figure 2.2. Two models showing the principle of superposition. If the state-space

model (which all four have the same coefficients) is linear, the left system and the right are equal.

The models developed and worked with at TDGT are highly nonlinear.

2.4.2 Linearization

It can be purposeful to se how the solution of a nonlinear system behaves “close to” a certain working point. If a solution to (2.1) is x0, u0, let y0= h(x0, u0). This

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x(t) − x0, ∆u(t) = u(t) − u0 and ∆y(t) = y(t) − y0. Then the approximation (2.3)

is valid.

d

dt∆x = A∆x + B∆u

∆y = C∆x + D∆u (2.3) where A, B, C and D are Jacobians to f (x, u) and h(x, u) with respect to x and u respectively, evaluated in (x0, u0).

Linearization is a useful tool, but there are restrictions [16].

• Linearization can only be used to study local properties close to the chosen working point. In many cases the behaviour around a stationary solution can be interesting, in particular since it is often desired that the system should reside around such a point.

• It is often difficult to quantitatively estimate how good an approximation the linearized solution is.

A question that arises in connection with linearization is: how big can the deviations ∆x and ∆u be before even these deviations cannot be considered linear? Normally the deviations used in sensitivity analysis are rather small, but in this thesis the deviations have been as big as 400% for the friction coefficients in the landing gear model.

The questions and restrictions around linearization show that experience is of importance when performing a linearization.

2.4.3 Robustness

Another influential property of a model is robustness. A robust system is such that its properties do not change more than expected if applied to a system slightly different from the mathematical one. Sensitivity analysis is the primary tool for studying the degree of robustness in a system [14].

Important to note is that because a system is robust with respect to certain parameters, this does not guarantee anything regarding the robustness of other parameters of the system.

2.4.4 Level of model validity

In the definition of a model, in 2.1, it is stated that a model has to be “good enough for the defined system or problem at hand”. To decide when a model is good enough, tests and evaluations are carried out to be sure the confidence in the model is sufficient and the model can be considered valid. Since no rules exist, engineers must use their experiences to determine if additional testing is warranted, but in practice a model’s level of validity is often a consequence of project constraints such as time, money and human resources.

Even when the results obtained from tests and evaluations are reasonable and the model seems to be correct, this might not be the case. If a model produces

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2.5 Modeling and simulations 13

an acceptable prediction, it could be because of: (a) reasonably correct input as-sumptions combined with a reasonably correct model structure; (b) compensating errors in the model input assumptions and a reasonably correct model structure; or (c) a combination of incorrect input assumptions combined with errors in the model structure, all of which compensate for each other [7].

In the following example a system equation is assumed to be as in (2.4).

f (x) = −u(x)

u(x) = −x (2.4)

The true system is described in equation (2.5).

f (x) = u(x)

u(x) = x (2.5)

This model is an example of the third case above, (c), where a combination of incorrect input assumptions combined with errors in the model structure, all of which compensate for each other. As long as the input has the wrong sign the modeler will not realize this model error, and the two errors cancel each other out. This short example shows that even when the simulation results seem to prove that the model is “good enough”, this might not be the case.

2.5 Modeling and simulations

2.5.1 Modeling and simulation tools

There are different types of simulation tools, two examples are Dymola [10] and the more commonly used Simulink [17].

Simulink is a signal-flow tool where the flow of information between components is causal (definition A.2). The different components are predefined as to what is input or output, with the signal relating them flowing in a specified direction. The equations are given on the form of ordinary differential equations (ODE), see (2.6). Simulink has a good support for data-flow and control-system modeling and is commonly used in software-intensive systems with discrete blocks [20].

˙x = f (u, x)

y = g(u, x) (2.6)

Dymola on the other hand is a power-port tool where the information is trans-ferred between components through power ports, connections that are non-causal, (see definition A.2). Dymola has a good support for physical modeling and the tool is suitable for continuous systems. The tool is based on differential-algebraic equations, DAEs, see (2.7).

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2.6 Uncertainties

2.6.1 Parameter uncertainties

When modeling a system there will most of the times be uncertainties that have to be addressed.

There are known knowns. There are things we know that we know. There are known unknowns. That is to say, there are things that we now know we don’t know. But there are also unknown unknowns. There are things we do not know we don’t know.

Former U.S. Secretary of Defense, Donald Rumsfeld February 12, 2002

Uncertainties can be distinguished as being either aleatory or epistemic.

Aleatory uncertainty: An aleatory uncertainty arises because of natural, un-predictable variations in the performance of the system under study. Aleatory uncertainties are also referred to as variability, irreducible uncertainties, inherent and stochastic uncertainties. The knowledge of experts cannot be expected to reduce aleatory uncertainty although their knowledge may be useful in quantify-ing the uncertainty. Thus, this type of uncertainty is sometimes referred to as irreducible uncertainty or external uncertainty [8].

Epistemic uncertainty: Epistemic uncertainty, or reducible and subjective un-certainties as it is also called, is due to a lack of knowledge about the behaviour of the system that is conceptually resolvable. The epistemic uncertainty can, in principle, be eliminated with sufficient study and, therefore, expert judgements may be useful in its reduction.

From a psychological point of view, epistemic (or internal) uncertainty reflects the possibility of errors in our general knowledge. For example, one believes that the population of city A is less than the population of the city B, but one is not sure of that [8].

Early in the concept phase, there are more epistemic uncertainties than aleatory uncertainties. During the refinement of the model, most epistemic uncertainties decrease and some epistemic parameters transform into aleatory uncertainties [21].

2.6.2 Model structure uncertainties

Epistemic uncertainties can also be connected to the model structure, that the system and its equations are not modeled with sufficient detail. While parameter uncertainty is linked to the physical parameters themselves, model structure un-certainty refers to lack of knowledge about the relationships between parameters and the underlying phenomenologies [3].

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2.7 Disturbances 15

Activity index

The effects of model structure uncertainties can be covered by comparison with test data. Another approach, not discussed in this thesis, is to use the so-called activity index, where the energy levels in a system are monitored, and parts with low activity are candidates for simplification and parts with high activity may need further elaboration [1].

2.7 Disturbances

A disturbing signal or disturbance is an external signal which we cannot choose or influence. Even though we cannot influence the disturbances, they often have impact on the system and therefore it is essential to have knowledge about their typical qualities [16]. The problem with disturbances will not be further investi-gated in this thesis but is assumed dealt with in the modeling process and is a part of the model uncertainties.

Known sources of disturbance - Measurable disturbances

The disturbance is often a well known physical quantity, and is measurable. In this case the disturbances can be modeled, and in many cases treated as an input, if it has been properly modeled.

Known sources of disturbance - Non measurable disturbances

Sometimes, even though the origin of a disturbance is known, it is not measurable. Take the case of a plane. Its motions are determined by the power of the engine, the force of gravity and by the forces exerted on the airplane from the air surrounding it. Some of these are known or measurable and can be treated as in the previous paragraph; others are known but not measurable and have to be treated as such.

Unknown sources of disturbances

A third case is when there are disturbances but where the possibility, time or energy is lacking to find out their physical causes. These effects can then be col-lected to one aggregated contribution which typically is added to the undisturbed output.

2.8 Automatic control engineering

When talking about a system or a system model in automatic control it is usually well defined what the inputs, outputs and the controlled quantity are. That is not always the case for the models discussed in this thesis. The signal can in different contexts be input and output and it is therefore difficult to use the same definitions that are used in automatic control.

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Still there are similarities. Perturbations causing disturbances have to be dealt with as well as uncertainties and discrepancies between the model and the reality.

2.9 Models used at TDGT

The models developed and used at TDGT today are complex, heterogeneous air-craft models including both equipment and the software controlling and monitoring the physical system. The systems include physical models (pumps, valves, gears, turbines, etc, i.e. fluids that moves) and software models (models of the code that controls the system) and also a mixture of continuous and discrete parts. When working with aircraft systems, in addition to the usual problems occuring during the modeling process, challenges such as the effects of g-forces and two-phased flow, i.e. mixing fuel with air, in fuel systems occur.

As an example of the complexity of the models in use, the fuel system model has 300 components with 226 state variables and more than 100 input and output sensors. The environmental control system is probably the most complex model with more than 100 component subroutines and five major feedbacks which has been in daily use for ten years. This model was the first for which Saab used a modern tool instead of the ordinary FORTRAN environment used earlier, and was modeled in the modeling tool Easy5. A migration from Easy5 to Dymola is in progress at TDGT and a part of the environmental control system modeled in Dymola can be seen in figure 2.3.

Figure 2.3. The second topmost layer of the environmental control system, with

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2.9 Models used at TDGT 17

All components in the systems have been physically modeled, such as valves, pipes, tanks, sensors, heat exchangers, ram air channels, water separators, cabin, on-board oxygen generating system compressors, turbines, etc.

The system context such as flight conditions (speed, altitude, humidity, tem-perature, etc.) are inputs to the model, as are the interfaces to other systems where energy is transferred into the system. Dynamic models based on physical differential equations have generally been used. Black-box models have been used for some equipments of minor interest such as sensors. Tables are only used for highly nonlinear equipment such as compressors and turbines [12].

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Chapter 3

Theory - Sensitivity Analysis

Sensitivity analysis (SA) is the study of how the variation in the output of a model (numerical or otherwise) can be apportioned, qualitatively or quantitatively, to different sources of variation, and of how the given model depends upon the information fed into it [19].

Since a model is always a simplification of the real system, the reliability of the model must be determined, and one way of doing so is using SA. By performing SA on a system, it can be determined which inputs influence a given output the most. The most important contributions to the uncertainty in a model can thus be determined, and this information can then be used as a guideline to determine the areas that need higher attention.

The Sensitivity analysis theory is presented in this chapter, starting with a description of global and local methods in section 3.1, followed by an account of the sensitivity measures Effective Influence Matrix, EIM, and Main Sensitivity Index, MSI, in section 3.2.

3.1 Global and local methods

There are different methods of performing sensitivity analysis. Local sensitivity stands for the local variability of the output by varying input variables one at a time near a given working point, which involves partial derivatives. The global sensitivity, however, stands for the global variability of the output over the entire range of the input variables and hence provides an overall view on the influence of inputs on the output. Using this variance-based SA the analysis of variance can be decomposed into increasing order terms, i.e. first-order terms (main effects) depending on a single variable, higher-order terms (interaction effects) depending on two or more variables [6].

Assuming that each parameter is computed in its nominal point and only one more point, the number of simulation runs needed when using a local method is

k + 1 for a system with k inputs, and 2k + 1 if a central difference approach is

used [21]. With a global method this number is 2k_{− 1 [19]. In other words a}

local method grows linearly with an increasing number of inputs, while a global

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20 Theory - Sensitivity Analysis

method grows exponentially. Thus, though global methods might work very well on small problems, the increase in required experiments soon becomes untenable with a growing number of inputs.

Furthermore, even a small model might not be a suitable candidate to global SA. If a model is stiff, has time constant of different magnitudes, simulation is often slow. Not only the number of simulations demanded but also the time needed to simulate influence the time it takes to perform a sensitivity analysis. Therefore a model which has few inputs but takes a long time to simulate is unfit to perform global sensitivity analysis on.

Seeing that the models used at TDGT are big and complex, the number of simulations needed for a global sensitivity analysis would be too big to be prac-tical. Consequently local sensitivity analysis methods have been chosen to be investigated in this thesis.

3.2 Local sensitivity measurements

To increase the understanding and the background of the sensitivity measures used in this thesis, a description of the notation is presented. The sensitivity measurements presented in this chapter are presented more thoroughly in [14].

Since different parameters, and at times also their sensitivities, might have values of different orders of magnitude it may be difficult to get a good overview of a system and how sensitive the variables actually are. In the landing gear experiment, the system characteristics vary between 1.5 · 10−6 _{and 1.9 · 10}7_{, a}

difference in magnitudes of 1013. Normalized variables are also used in order to facilitate comparison between different units as well as making it possible to use addition, e.g. when wanting to aggregate variables into one single variable. Thus, in order to make it easier to get an overview of the sensitivities, normalised dimensionless sensitivities are introduced.

Sometimes in SA, the range of variation is taken as identical for all the variables, often a few percent of the nominal value [19]. The approach in this thesis is not to have the same relative deviation for all the parameters, but to be able to choose these freely.

The uncertainties of system characteristics, y, can be expressed as a function of two vectors, xdand xu, design parameters and uncertainty parameters respectively,

see equation (3.1). An example of a design parameter could be the choice of fluid in a system whereas an uncertainty parameter could be the viscosity fo that fluid, which is a function of the temperature (among other things).

y = f (xd, xu) (3.1)

f is a nonlinear function but if it can be linearized around a nominal point, this

will result in the equation (3.2). For some reflections on the correctness of the assumption of linearity, see section 6.3.

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3.2 Local sensitivity measurements 21

hence

∆y = A∆xd+ B∆xu (3.3)

A and B are Jacobians with elements:

aij = δfi(xd, xu) δxd,j = δyi δxd,j (3.4) bij = δfi(xd, xu) δxu,j = δyi δxu,j (3.5)

where the aij element in the A-matrix describes the sensitivity on the i:th system

characteristic by the j:th desing parameter index. The bij element in the B-matrix

describes the sensitivity on the i:th system characteristic by the j:th uncertainty parameter index. With a high value of aij, a change in the j:th parameter will

result in a relatively bigger change in the i:th system characteristic than with a smaller value of aij, see figure 3.1.

0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 18 20 x + ∆_x y + ∆y

Figure 3.1. aij and bij describe the relation between ∆x and ∆y. The values in the

graph are examples with a normal distribution. In this case the aij or bij is linear and

equals 2. With a nominal x-value of x = 5 and a deviation of ∆x = 0.5 this would lead to a ∆y = 1.

The normalized sensitivities become:

b0,ij= xuj

yi

δyi

δxuj

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22 Theory - Sensitivity Analysis

In this way, a non-dimensional value is obtained that indicates by how many percent a certain system characteristic changes when a system parameter is changed by one per cent. By studying the effect of these uncertainties valuable information about parts in the design can be found, and a sensitivity matrix can be estab-lished. With this matrix it is easy to see which areas need higher attention to reduce uncertainty.

In this thesis two sensitivity measures have been investigated, and these are presented below.

3.2.1 Effective Influence Matrix, EIM

Valuable information can be collected by looking at the actual uncertainties in pa-rameters. On condition that the uncertainties are small compared to the nominal values, the variance in the system characteristics can be calculated as:

Vy,i=

n

X

j=1

b2ijVx,j (3.7)

Here Vx,j is the variance in the parameters and Vy,i is the variance in the system

characteristics. The standard deviation can then be calculated as:

σy,i=pVy,i= v u u t n X j=1 b2 ijVx,j = v u u t n X j=1 b2 ijσx,j2 (3.8)

and the normalized (non-dimensional) standard deviation becomes:

σ0,y,i= σy,i yi = 1 yi v u u t n X j=1 b2 ijσx,j2 = 1 yi v u u t n X j=1 y_i xj b0,ij 2 (xjσ0,x,j)2 = v u u t n X j=1 b20,ijσ0,x,j2 (3.9)

The influence from removing an uncertainty altogether can then be calculated. The difference in variance in system characteristics by removing one uncertainty

Vx,j is:

∆Vy,ij = Vy,i− Vy,ij∗ (3.10)

Here V∗

y,ij is the variance for the system characteristic, but with the variance of

the j:th parameter set to zero:

V∗

y,ij= Vy,i,Vx,j=0 (3.11)

For a linear system or a nonlinear system with small variations, this becomes:

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3.2 Local sensitivity measurements 23

The change in deviation can then be expressed by:

∆σy,ij = σy,i− q Vy,i− b2ijVx,j = σy,i− q σ2 y,i− b2ijVx,j = σy,i 1 − s 1 − b2ij σx,j2 σy,i2 ! (3.13) or in non-dimensional form: ∆σ0,y,ij= σ0,y,i 1 − s 1 − b20,ij σ20,x,j σ2_0,y,i ! (3.14)

The EIM states how much smaller the deviation will become if the coupling between this system parameter and this certain system characteristic was perfect. If a variation in the j:th parameter has a big impact on the i:th characteristic variation, this will lead to a bigger ∆σ0,y,ij than if the j:th parameter has a small

impact. Due to the non-linear form of (3.13) there are often very few significant elements in the Effective Influence Matrix; a large deviation in one parameter quickly shadows the influence of other parameters.

3.2.2 Main Sensitivity Index, MSI

The Main sensitivity index SM SI,i,j, is a measure of influence for uncertainties,

defined as the ratio between the contribution to the total variance in a system characteristic i by an uncertain parameter index j, Vy,i − Vy,ij∗ , and that total

variance in a system characteristic index i, Vy,i.

SM SI,i,j= Vy,i− Vy,ij∗ Vy,i = 1 −V ∗ y,ij Vy,i (3.15)

Under the assumption of small uncertainties the behaviour is linear and can be written as: SM SI,i,j= b2ijVx,j Vy,i =b 2 ijσ2x,j σ2 y,i (3.16)

The results can be easily read in a MSI matrix, as the portion of influence a given parameter has on the system parameters. In the MSI matrix the values are normalized so the row sum is always one [14].

The MSI only provides the first-order interaction effects, but there is also a sensitivity index closely linked to the MSI, the Total Sensitivity Index, TSI. The TSI include terms of higher orders, rapidly increasing the number of simulation runs needed.

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Chapter 4

Experiments

In order to examine the sensitivity measures described, two tests have been made. One on a laboratory equipment including a hydraulic servo, described in sec-tion 4.1, and one on a conceptual landing gear model of the Gripen aircraft (from now referred to as the landing gear), described in section 4.2.

4.1 Hydraulic servo experiment

The laboratory equipment in the hydraulic servo experiment consists of an I-beam that can seesaw around its center. The position of the beam is controlled by a hydraulic servo, see figure 4.1 for a sketch and figure 4.2 for a picture. Weights can be added to the ends of the beam to increase the inertia.

This model could represent the movements of beams and other equipments in a mechanical system. A hydraulic servo is a part of the landing gear being investigated later in this thesis. The aim of this experiment is to test the local sensitivity analysis method on a quite small and well defined problem. The system is easily handled and small enough to get the general view of.

Figure 4.1. Laboratory equipment for the hydraulic servo experiment, [9].

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26 Experiments

Figure 4.2. Laboratory equipment for hydraulic servo experiment, with added weights

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4.1 Hydraulic servo experiment 27

4.1.1 Laboratory equipment

The functioning of the laboratory equpiment is described below, with the numbers referring to figure 4.1.

1. PC With dSPACE and a graphical user interface (control desk) to monitor and change the controlling parameters.

2. dSPACE measurement and control system.

3. Servo valve MOOG D661 (140 l/min), two stage flapper valve with mechan-ical feedback. Electrmechan-ical feedback of the valve spool position.

4. Pressure sensor for measuring the cylinder pressures, p1 and p2 and the

system pressure ps.

5. Symmetric servo cylinder. The cylinder has a built-in position sensor (po-tentiometer sensor).

6. Pressure relief valve to set the system pressure ps.

A pump and tubes couples oil to the system, and a pressure relief valve (6 in figure 4.1) sets the system pressure ps. The tubes are connected to the servo

valve (3) which in turn is connected to the hydraulic cylinder (5), determining the position xp of the beam. The valve is a 4-way servo valve and is controlled with a

voltage signal from the controller. p1, p2(4), psand xp(the built-in potentiometer

in 5) are measured and sent to a computer via a dSPACE measurement and control system (2), and a valve signal is computed and sent to the servo valve. A PI (pro-portional - integral) controller is implemented in Matlab/Simulink in a connected computer (1) and communicates with the hydraulic system via dSPACE.

The graphical interface with the slides to adjust the P- and I-element param-eters as well as the choice of wave form (with amplitude and frequency settings) and the resulting outputs from the system is presented in figure 4.3.

4.1.2 HOPSAN

HOPSAN, created at Linköping University in 1977, is a simulation tool for hy-draulic power systems but has also been adopted for other domains such as elec-tric power, flight dynamics and vehicle dynamics. The existence of a library of component models of for example valves, machines and lines makes it possible to study complex load dynamics and wave propagation in long lines [15].

The laboratory experiment used in experiment one can be seen modeled in figure 4.4.

4.1.3 Co-simulation with HOPSAN and Excel

HOPSAN can not only simulate systems on its own but can also communicate with other tools during simulation. The program can collaborate with Matlab/Simulink,

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28 Experiments

Figure 4.3. The graphical interface of the equipment in the hydraulic servo experiment.

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4.2 Landing gear experiment 29

Excel and Visual Basic in different ways, and in this thesis the possibility to co-simulate with Excel has been made use of. Through Excel most Windows programs can communicate with HOPSAN, for further information read the user’s guide [15]. Using co-simulation it is possible to list the variables in the HOPSAN model in Excel and use Excel to change the variables in HOPSAN. This has been employed when calculating the sensitivity measures defined in section 3.2. By co-simulation the variables used in HOPSAN can be used by Excel, and Excel can add the prede-fined ∆x to each system parameter and calculate the EIM:s and MSI:s connected.

4.2 Landing gear experiment

The Gripen has three landing gears; two main landing gears, one attached to each wing, as well as a nose gear. The main landing gear examined in this thesis, consists of a main leg, the integrated wheel fork and plunger tube, to which the wheel is attached, a fold stay including a lock mechanism and a hydraulic circuit extending the landing gear, see figure 4.5. Figure 4.6 shows a picture of the Gripen aircraft during flight.

4.2.1 The landing gear

The landing gear has been simulated in the case of a normal unfolding with the help of the hydraulic cylinder as well as an emergency unfolding. In the extension case, an hydraulic cylinder together with the air loads will help the landing gear unfold. In the case of an emergency unfolding, a catch will unlock a cover and the landing gear will fall down, and gravitational forces and the air load will unfold it, that is to say the hydraulic system will not be in use.

This experiment is to examine the influence of different frictions on the un-folding of the landing gear - mainly to examine whether or not it will extend fully and reach the end position when in emergency state, or if it will be suspended half way. The time to reach full extension has also been investigated, as well as the pressure in the two tanks in the hydraulic cylinder and the flow of hydraulic oil to and from the landing gear.

In this experiment no physical testing will be performed, only simulations.

4.2.2 Dymola

The landing gear is modeled in Dymola. Dymola is a component based tool for modeling and simulation and is developed by Dynasim AB, a company based in Lund. The ideas behind Dymola were originally developed by Hilding Elmqvist as a part of his Ph.D. thesis, which he attained at the Department of Automatic Control, Lund Institute of Technology in 1978. Dymola uses the generic modeling language Modelica, which is an object-oriented, declarative modeling language. Modelica is component-oriented and is used primarily for complex systems such as mechanical, electrical, hydraulic, thermal, control and electric power components, and is also capable of handling systems with two or more of these types.

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30 Experiments

Figure 4.5. A drawing of the Gripen main landing gear.

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32 Experiments

Initial value problem

Many simulation programs need a period of time to initialize and find the right stationary states, and therefore you cannot order an event until a certain period of time into the simulation, in order to let the system stabilize to a stationary point. This is not needed when using Dymola as it starts by solving an initial value problem. Still this requires correct initial values if initial oscillations are to be avoided. One way to do this is to run a simulation for an appropriate time, save these final values obtained and use these as initial values.

Communication between Excel and Dymola

With the sensitivity analysis code written in Visual Basic for Application (VBA) in Excel and the model being modeled in Dymola, these programs need a means for communication. This is done using Dynamic Data Exchange (DDE), a technology for communication between different applications.

The user fills in the system parameter values and their deviations in Excel, and the simulations will be executed as a batch. Excel/VBA will set the parameters and send these to Dymola, which executes the simulations and returns the values of the predefined system characteristics, and the local sensitivity analysis result is calculated and presented in Excel. A simplified flow chart on the communication between Excel and Dymola can be found in figure 4.8.

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The maximum length of a DDE command is 255 characters, which is a rather small number when the information demanded by Dymola includes model name, start values and final values. The modeler might also want to state the simulation time and method, quickly resulting in more than 255 characters. To avoid the problem of a commando string being too long, the system characteristics used in the simulations in this thesis have been renamed in Dymola. This was done directly in the Dymola model, but when applying the method on bigger models and the analysis being used in everyday life it is not a tenable way to do it. A suggestion on how to solve this problem is to write a part of the code in a Dymola script, but this entails other problems, see further in Appendix C.

The Excel interface where the system parameters are filled in is shown in figure 4.9 and the system characteristics in figure 4.10. For a user’s guide on how to set the different values, see Appendix B.

Figure 4.9. System parameters used in the landing gear sensitivity analysis.

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Chapter 5

Results - Hydraulic servo

experiment

This chapter starts with a comparison between the physical model and the HOP-SAN model used for simulation, in section 5.1. The system parameters and char-acteristics used in the sensitivity analysis are presented in section 5.2, followed by the results for a step input and a square wave input in sections 5.3 and 5.4 respectively. Section 5.5 presents some observations made in this experiment.

5.1 Comparison between physical system and

HOPSAN model

One way to compare is to utilize the properties of control theory. Four properties have been looked closer at. These are overshoot, rise time, steady-state error and the time constant. The overshoot is defined as M = (ymax−yf)/yf where yf is the

steady-state value and ymaxis the maximum value of the signal. The overshoot is

often given in percents. The rise time, Tr is the time it takes for the signal to go

from 0.1yf to 0.9yf. The steady-state error is defined as the difference between

the steady-state value and the reference signal. These can be seen in figure 5.1. A time constant is used to see in which time scale the output approaches the steady-state value yf.

The results of these tests are shown in table 5.1. More detailed pictures of a step response for the physical system as well as the one modeled in HOPSAN are shown in figures 5.2 and 5.3 respectively.

5.1.1 Differences between the model and reality

The hydraulic system has been at least partially calibrated, but it was unknown when and how this was done. Since the purpose of this thesis is not to perfect a model for the experiment equipment described, the model has been adapted to

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36 Results - Hydraulic servo experiment 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time [s] steady−state error overshoot y f 0.1 y_f 0.9 y f T r

Figure 5.1. A step response example to clarify the terms: overshoot, rise time and

steady state error. The overshoot marked in the figure should be divided by yf to obtain

a ersult in percents.

Table 5.1. Comparison between the control theoretic properties of the hydraulic servo

system.

Real system HOPSAN model

Time constant τ [s] 0.11 0.12 Rise time[s] 0.12 0.14 Steady-state error - 0 Overshoot[%] 2.5 4.0 0 2 4 6 8 10 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 "Model Root"/"Xp f ilt"/"Out1" "Model Root"/"Xpref" x pos x pos,ref

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5.2 System parameters and characteristics 37

Figure 5.3. Step response for the HOPSAN model, plotted the first 20 seconds and the

first second respectively.

accord with the reality. This was done by changing the system gain, resulting in a system model that corresponds sufficiently with the real system.

5.2 System parameters and characteristics

The system characteristics chosen to investigate are: the spool position and the pressure levels in the piston, see table 5.2. The Relative model uncertainty, column 5, is an estimation of the model structure uncertainty, given in percent. This is to be able to compare the influence of uncertainties in the parameters with the overall uncertainty in the model structure; is it really worth refining the model if the uncertainties in e.g. the weight of the plane is much bigger than the model structure uncertainty?

Table 5.2. System characteristics investigated in sensitivity analysis.

System characteristics Name Value Unit Rel. model unc. Pressure node A P_SERVAL_1_NA 3104729 Pa 0.005 Pressure node B P_SERVAL_1_NB 2977669 Pa 0.005 Spool position XS_XSENSE_1 0.141110 m 0.01 Spool position X_MLOADC_1_NX2 0.141110 m 0.005

Once the system characteristics to investigate are decided, the parameters that will or might influence these are elected. These can e.g. be known or be suspected to have an influence on the system characteristics, or could be chosen to secure they do not affect the same. The system parameters chosen are: oil density and flow coefficient Cv or Cq in the valve, the bulk modulus of the oil and the piston

areas a1and a2, the load mass of the bulk and the system gain, see table 5.3. The

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38 Results - Hydraulic servo experiment

Table 5.3. System parameters used in sensitivity analysis.

System group Name Value Variability Unit Oil density RHO_SERVAL_1 870 100 kg/m3 Flow coefficient CQ 0.67 0.1

Bulk modulus of oil BETAE_PISTON_1 8.00E+08 4.00E+07 Pa Piston area A1 7.60E-04 1.00E-04 m2 Piston area A2 7.60E-04 1.00E-04 m2

Mass ML_MLOADC_1 387 100 kg

Gain: Multiplier K_GAIN_1 0.02 0.002 Amplitude input ystep_Pulsetraininput_1 0.08 0.02

5.3 Step response

The Effective Influence Matrix, see 3.2.1, performed on the HOPSAN model results in a matrix with rather small elements, as can be seen in figure 5.4.

Figure 5.4. EIM of a step response.

When starting out the process of deciding whether to go forward with the model as it is or if it has to be worked with more it is useful to look at the normalized deviations of the system characteristics. In this case they are between one and ten percent. If these are bigger than a threshold value, further investigation might be needed.

Since the values in the EIM and the MSI matrix are based on the same in-formation, these are different ways of presenting the same information. The EIM states how much smaller the deviation will become if the coupling between this system parameter and this certain system characteristic was perfect. With the values being rather small, it might not be economically defensible to continue the refinement of the model.

The Main Sensitivity Index matrix (see 3.2.2) can be seen in figure 5.5. At a first glance it might look as if the piston areas A1 and A2 have a big impact on the pressures P _SERV AL_1_N A and P _SERV AL_1_N B, with element values between 42 and 57 percent, but with a closer look it will be shown that these connections are not as bad as it could seem. Since the sum of the elements

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5.3 Step response 39

in each row is always one in the MSI matrix, these are only the internal relations of the system parameters. If, as in this case, A1 and A2 are the only parameters influencing the pressures A and B, these elements will have rather high values. It might mean that these connections should be further investigated, or just that this system characteristic is affected by these parameters and these parameters only.

To decide which case it is, one has to look at the normalized deviations, on condition that the deviations chosen are big enough. If the normalized deviations are big enough, the model might be in need of more work if this deviation seems bigger than expected, otherwise it is probably okay.

Figure 5.5. MSI of a step response.

One of the benefits with the MSI matrix is its lucidity. It is easy to see which parameters that have the biggest influence on each system characteristic, and with the percentage presentation it is not necessary to have a deeper understanding of the system being analyzed. At the same time, this is also a drawback, since by the conversion into percentages some of the connection to the real system and the understanding of the same is lost.

5.3.1 Step response with new system characteristics and

some new parameters

Another sensitivity analysis has been performed on the model with a step input. In this experiment, the characteristics chosen to analyse are: the integral of the square of the fault, Rtend

t=0 (y − yref)2dt, and the integral of the absolute value of

the fault,Rtend

t=0 |(y − yref)| dt, see table 5.4.

Two system parameters are added as well, the viscous friction coefficient Bp

and the coulomb friction force Fc, as can be seen in table 5.5.

The updated HOPSAN model is shown in figure 5.6.

Table 5.4. Additional system characteristics investigated in sensitivity analysis.

System characteristics Name Value Unit Rel. model unc. Integrated (absolute(fault)) y_I1Filter_1 0.024 Pa 0.01 Integrated ((fault)ˆ2) y_I1Filter_3 9.5E-4 Pa 0.01

Local Sensitivity Analysis of Nonlinear Models - Applied to Aircraft Vehicle Systems