Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch ...

Literature Survey

Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch-Regulated Wind Turbine S.K. Zegeye July 16, 2007

Delft Center for Systems and Control

Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch-Regulated Wind Turbine Literature Survey

S.K. Zegeye July 16, 2007

Faculty of Mechanical Engineering

·

Delft University of Technology

Delft University of Technology

c Delft Center for Systems and Control Copyright All rights reserved.

Preface

This literature survey proceeds the thesis project and report entitled as ”Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch-Regulated Wind Turbine”. The project is supervised by

Prof. dr. ir. M. Verhaegen(DCSC) Ir. J.W. van Wingerden (DCSC)

Delft University of Technology July 16, 2007

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Preface

Literature survey

Table of Contents

Preface

iii

Acronyms

1

1 Introduction

3

1-1 Goal and Motivation of this literature survey . . . . . . . . . . . . . . . . . . . .

4

1-2 Outline of the survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Wind Turbines

5

2-1 Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2-2 Modeling of Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2-2-1

Mathematical model of Wind turbines . . . . . . . . . . . . . . . . . . .

7

2-3 Challenges in Wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2-4 Conclusions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

3 Linear Parameter Varying Systems

13

3-1 Overview on LPV systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3-2 LPV modeling of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . .

15

3-2-1

Taylor-Series Expansion Linearization Method . . . . . . . . . . . . . . .

15

3-2-2

Function Substitution Method . . . . . . . . . . . . . . . . . . . . . . .

18

3-2-3

State Transformation Method . . . . . . . . . . . . . . . . . . . . . . . .

21

3-3 Conclusions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

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Table of Contents

4 Identification of LPV systems

23

4-1 Overview of System Identification . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4-2 Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4-2-1

LTI Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . .

25

4-2-2

LPV Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . .

28

4-3 Conclusions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

5 Model Validation

33

5-1 Overview on Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5-2 LMI approach to LPV model (in)validation . . . . . . . . . . . . . . . . . . . . .

35

5-2-1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

5-2-2

Problem statement and results . . . . . . . . . . . . . . . . . . . . . . .

36

5-3 Conclusion and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

6 Concluding Remarks Bibliography A Formal MSc-Project Agreement

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Literature survey

List of Figures

2-1 Fixed speed wind turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2-2 Variable speed wind turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2-3 Wind turbine modelled as a set of four interacting modules. . . . . . . . . . . .

7

2-4 A small annular section rotor blade. . . . . . . . . . . . . . . . . . . . . . . . .

8

2-5 Mechanical dynamics of wind turbine. . . . . . . . . . . . . . . . . . . . . . . .

9

3-1 Block diagram representation of ULFT Fu (M, ∆(p)) . . . . . . . . . . . . . . .

14

5-1 General scheme in model development and (in)validation . . . . . . . . . . . . .

34

5-2 LPV model (in)validation setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

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List of Figures

Literature survey

Acronyms

BMI Bilinear Matrix Inequality DCSC Delft Center for Systems and Control DUT Delft University of Technology KWh Kilo Watt hour LFT Linear Fractional Transformation LMI Linear Matrix Inequality LPV Linear Parameter Varying LTI Linear Time Invariant LTV Linear Time Varying MIMO Multi Input Multi Output SISO Single Input Single Output VAF Variance Accounted For WECS Wind Energy Conversion System

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Acronyms

Literature survey

Chapter 1 Introduction

The concern for environmental pollution, the increasing demand of electricity, the limitation in fossil fuel resources, political and economic concern draw the attention of governments and organizations to seek sustainable energy sources. Wind energy is one of the renewable and non-pollutant energy sources. It has the potential of supplying 2 × 1013 W [29] of power. Nevertheless, the contribution of wind energy is still very small relative to the electric energy consumption of the world. Many of the reasons account for the energy conversion system. Currently wind turbines are used for the conversion of wind energy to electric energy. Recently wind energy has received a lot of concern. Wind energy industries and researchers are undergoing research to increase the amount of power that can be obtained by reducing the fatigue and load of wind turbines. Since the maintenance cost of wind turbines is one of the main reasons for the increase of per Kilo Watt hour (KWh) cost of electricity generated by wind turbines, it draws the attention of control engineers to design an optimal robust controller to increase the efficiency and robustness of the system. However, this is not as simple as it looks. Control techniques are well developed for linear systems. But since the dynamics of wind turbines are highly nonlinear, complex, time varying, and coupled, the task in designing controller become challenging. Then, it is vital to find an approximate model of the system which can suite it self for the available controller design techniques. Linear Parameter Varying (LPV) modeling is one of the techniques which can be used to model a system as parameter dependent Linear Time Invariant (LTI) systems. This modeling technique better approximates nonlinear systems than LTI modeling technique. An example of LPV modeling technique as applied to wind turbines is discussed in §3-2-1. This triggers that LPV modeling of wind turbines is a milestone in designing optimal and robust wind turbines.

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Introduction

In this literature survey the dynamics of wind turbines is discussed. The challenges with respect to this system are surveyed. The potential of LPV modeling techniques are explored too.

1-1

Goal and Motivation of this literature survey

Literature survey, in general, serves a two fold goal. Initially it provides clear understanding of the system understudy. That is to say, it aids in unfolding the areas so far explored and areas under research. Secondly, it helps in defining the problems in the system. It provides a clear explanation of the challenges that need further investigation and study. Obtaining a suitable model and designing controllers is common practice for control engineers. A control engineer needs to have thorough understanding of a system, so that he/she can model it in the way it suits into the controller design algorithms. The aim of this literature survey is to develop basic knowledge on wind turbines and LPV modeling techniques which helps in a completion of a thesis-project titled as: Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch-Regulated Wind Turbine. The title captures the four main topics of the report, namely: basic wind turbine theory, LPV systems, LPV identification and model validation. The choice of LPV modeling of the system is motivated from the fact that wind turbine dynamics is time varying and highly complex and nonlinear system. LPV modeling techniques extract the nonlinearity of systems into a time-varying scheduling vector. Thus, LPV modeling can be used to handle the nonlinearity of systems in a way which suits for controller design and has better approximation properties than LTI modeling.

1-2

Outline of the survey

The survey briefly explores the theory on wind turbines, and potential modeling techniques; particularly, LPV modeling of systems. Chapter 2 describes the working and dynamics of wind turbines. and the existing challenges in the system. In Chapter 3 an overview is given of the current state of the art literature in the area of analytical LPV modeling. In Chapter 4 techniques in developing a model of a system from real time data is addressed. Since models has to be validated with respect the real system, Chapter 5 addresses model validation technique that can be applied. Finally, Chapter 6 winds up the literature survey with concluding remarks and the problem statement for my M.Sc. thesis.

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Literature survey

Chapter 2 Wind Turbines

One of the renewable energy resources is wind. Wind energy propels boats, drives windmills, and generates electricity in wind turbines. Wind energy has grown during the last decades and nowadays it is the most competitive form of renewable energy [11]. The three main reasons for the growing concern in wind energy are: limited fossil fuel resources and their impact on the environment, constantly increasing demand in electricity, and political dependence on oil exploring countries [14]. These days, in order to harvest the vast energy of wind in the form of electric power, wind turbines are used. Thus, study of the background theory of wind turbines is crucial. However, all the details of wind turbines can not be covered in this single chapter. Here in the sequel some of the basic points in wind energy §2-1, modeling of wind turbines §2-2, and the challenges in wind industry §2-3 will be addressed. Finally the chapter will end with conclusions and results §2-4.

2-1

Wind Energy

Wind is the movements of air masses in the atmosphere. The movements are primarily caused by the difference in the temperature within the atmosphere, which is the result of unequal heating of the crust of the earth by the sun [14]. Consequently, the behavior of wind depends on the general climate of a region, the physical geography of the locality, the surface condition of the terrain of the site, and other factors [14]. Hence, the random geographic nature of the earth and terrain causes the wind to be random both in direction and speed. This creates a major challenge in harvesting wind energy. Meteorologists estimate that about 1% of the incoming solar radiation is converted to wind energy. However, according to World Meteorological Organization it is only possible to extract about 1.67% of the wind energy available in the atmosphere due to limitations in height and accessibility. Nevertheless, the amount of energy that can be extracted Literature survey

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Wind Turbines

(≈ 2 × 1013 W ) is large enough relative to the current world electricity consumption (i.e. an average of 0.15 × 1013 ) [14]. However, the contribution of wind energy to the present electricity consumption is very small. Since worldwide demand for electricity continues to rise, renewable energy growth rate has increased in recent years [29]. Thus, wind energy has grown widely in the last decades. However, wind energy is not cost effective. In consequence, the development of new technology will be crucial that the wind energy penetrates into electricity market successfully [11]. The knowledge of the undisturbed wind velocity Vun is essential for the wind energy conversion systems to impose loads with safety and be cost effective. Implementation of advanced control systems is considered as a promising way to improve wind turbine conversion systems [11].

2-2

Modeling of Wind Turbines

Wind turbines are systems which convert the kinetic energy of atmospheric air to electrical energy. Unlike other energy conversion systems (for example: nuclear, fossil fuel energy conversion system, etc...)the input(wind) to wind turbines are not controllable and are stochastic. Effective wind speed acting on the whole turbine rotor is a fictitious quantity and is thus not measurable nor available for control [11]. Wind turbines, are then, susceptible to fatigue and physical damage, which increase the per Kilo Watt hour (KWh) cost of the electricity generated.

Rotor

There are mainly two types of wind turbines, namely: fixed speed and variable speed wind turbines. In fixed speed wind turbines the generator is directly connected to the grid (see Fig. 2-1) and the speed of the rotor is maintained nearly constant, whereas in variable speed wind turbines an electronic power conversion system is connected between the generator and the grid (see Fig. 2-2) while the rotor speed can vary. Thus the load on the rotor of variable speed wind turbines is reduced.

Generator

wind

Drive train

Pitch angle

Actuator

Power to grid

Desired Controller pitch

Rated power

Figure 2-1: Fixed speed wind turbine

Any wind turbine can be considered as a set of bilaterally coupled modules. Where the Modules are Wind, Aerodynamic, Mechanical, Electrical, and Controller (In the case of offshore wind turbines Hydrodynamic and Wave are also included) [14]. The block diagram in Fig. 2-3 shows the interconnection of the modules of onshore wind turbine. Molenaar [14] has well described all the modules. The aerodynamic module converts the input wind field S.K. Zegeye

Literature survey

Rotor

2-2 Modeling of Wind Turbines

wind

7

Generator

Pitch angle

Tge

Actuator

Desired pitch

Generator speed

Drive train

Power Power to Electronics grid

Controller

Rated power

Figure 2-2: Variable speed wind turbine

to aerodynamic forces, which in turn are inputs into the mechanical module where they are converted to velocities. Note that the mechanical part of the generator are included in the mechanical module. Finally the electrical module converts the mechanical power into electrical power. The controller does act on both the aerodynamic and electrical module by computing set-points pitch angle θ, of the rotor blades and generator toque Tge , for the generated respectively. Since the literature survey concentrates on the aerodynamic and mechanical module Pge ,ref

Controller

T

Wind

Vun

Aerodynamic

Mechanical

Tge

Pge

Electrical

Figure 2-3: Wind turbine modelled as a set of four interacting modules.

of the wind turbine, all the modules of the wind turbine are not discussed here. An interested reader is referred to PhD thesis of Molenaar [14]. In the subsection below the equations governing wind turbines are described, giving due attention to the aerodynamic and mechanical modules.

2-2-1

Mathematical model of Wind turbines

The only external input to wind turbines is the undisturbed unknown wind velocity Vun . The interaction between the aerodynamics and the structural dynamics takes place via the blade movements and the aerodynamic forces. The aerodynamic forces are changed to rotational motion in the mechanical module of the wind turbine model given in Fig. 2-3. The speed of free stream of air Vun , is reduced as flowing towards the rotor plane, due to the increase in the dynamic air pressure. The fractional decrease of the wind velocity between the free air steam and the rotor plane is defined as axial interference factor, a = Vr /Vun , where Vr is the reduced wind speed velocity. Thus the wind velocity at the rotor plane is given by: Vo = Vun (1 − a) Literature survey

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Wind Turbines

Considering an annular section of the rotor with a small section of length ∆r, at a distance r from the center of rotation as shown in Fig. 2-4, the aerodynamic forces responsible for the rotation and flapping of the rotor blades are obtained as: dFto = dLcos(φ) + dDsin(φ)

(2-2)

dFQ = dLsin(φ) − dDcos(φ)

(2-3)

Where dFto is the flapping trust force, dFQ is the aerodynamic force generating rotational torque, and φ is the incident angle of the average wind velocity Vw . φ is the sum of the angle of attack α, and the pitch angle θ. The lift dL and drag dD forces are dependent on lift coefficient Cl (α, λ) and drag coefficient Cd (α, λ) respectively. Where λ = RΩ Vw is tip to wind speed ratio. The lift, dL, and drag, dD, forces are given in Eq. (2-4) and Eq. (2-5) respectively. 1 2 ρV Cl (α, λ)c∆r 2 w 1 dD = ρVw2 Cd (α, λ)c∆r 2 dL =

(2-4) (2-5)

with air density ρ, and cord length c. axis of rotation

dL plane of rotation

I

T

dD

D Vw r: ro

V

c 4

p

V o x r E

3c 4

Figure 2-4: A small annular section rotor blade.

From Fig. 2-5, the three main mechanical motion of the wind turbine can be described as, rotational motion of the turbine blades about the x−axis, the flapping motion of the blades about the y−axis and the vibration of the tower along the x−axis. The three equations governing these motions of the wind turbine model can be approximated as second order linear equation. Flapping motion of the blades is affected by gravity and the centrifugal force. Hence the equation describing the flapping of the blades will be ¨ + (mbl lgsin(ϕ) + kbl + Jbl Ωro (t)2 )β(t) = Mbl (t) Jbl β(t)

(2-6)

where Jbl is the moment of inertia of the blade about y−axis, mbl is the mass of the blade, l is the length between the center of mass of the blade to the axis of rotation, g is gravitational constant, kbl stiffness of the blade, Mbl is the flapping torque of the blade and Ωro is the rotational speed of the rotor. The torque, Mbl , of the blades can be obtained from Eq. (2-2) by multiplying the perpendicular component of dFto to the blades, by r and then integrating over the whole length of the blades. S.K. Zegeye

Literature survey

2-2 Modeling of Wind Turbines

E

Tro

9

M

k ro J ro

Tro

d ro Q J ge

Vun (1 a:) ro

kbl

Tge

d to

y

x

kto

y

z

z

Figure 2-5: Mechanical dynamics of wind turbine.

Since the Drag force is small relative to the lift force, it can be neglected and the integral expression becomes Z R Z R 1 2 dFto (t)cos(β(t))r ≈ ρVw Cl (α, λ)cos(φ(t))β(t)crdr (2-7) Mbl (t) = 0 0 2 where R is the radius of the rotor blades. By considering the mass of the nacelle, the rotor, and part of the tower as a point mass mto (tower equivalent mass), the vibration (nodding) of the tower at the nacelle position can be expressed as second order differential equation. The equation describing the motion is given in Eq. (2-8). Z R mto x ¨(t) + dto x(t) ˙ + kto x(t) = B dFto (t) (2-8) 0

where B is the number of blades, dto is the damping of the tower, and kto is the stiffness of the tower. From the dashed box of Fig. 2-5, let define ε˙ = Ωro − νΩge , as the twist velocity of the drive train, such that the rotor rotational dynamics can be described as ˙ ro (t) + dro ε(t) Jro Ω ˙ + kro ε(t) = Tro (t)

(2-9)

where Ωro is the angular speed of the rotor about the x−axis, Jro is the moment of inertia of the rotors about the x−axis, dro and kro are the damping and stiffness of the drive train respectively, and the torque, Tro , is given by Z R Z B R Tro (t) = rdFQ ≈ ρVw2 Cl (α, λ)sin(φ)crdr (2-10) 2 0 0 Literature survey

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Wind Turbines

with dFQ given in Eq. (2-3), and the effect of drag is neglected. The relation between the generator torque, Tge and the rotor speed, Ωro can be written as ˙ ge = ν(dro ε˙ + kro ε) − Tge Jge Ω where ν = nΩro /nΩge is the gear ratio. On substitution of Ω˙ ge = (2-9) into Eq. (2-11) we get a relation between Tge and Tro as:

(2-11) ˙ ro −¨ Ω ε ν ,

˙ ro from Eq. and Ω

Jro Jge Jge Jro ε¨(t) + dro ε(t) ˙ + kro ε(t) = Tro (t) + νTge (t) Jge + ν 2 Jro Jge + ν 2 Jro Jge + ν 2 Jro

(2-12)

Since the power generated, Pe , is measurable, the torque, Tge , and the angular velocity, Ωge , of the generator can be related with the equation Tge (t) =

Pe (t) ηΩge (t)

(2-13)

Where η is the efficiency of generator.

2-3

Challenges in Wind turbines

In §2-2 the equations describing wind turbine has been discussed. The formulas described in Eqs. (2-6), (2-8), (2-9), and (2-12) are simplified equations, which are obtained by making some assumptions (like the flap angle β(t) is small, no effect of feathering and lead-lag motion of the blades). In spite of the assumptions and simplifications of the dynamic equations describing the wind turbine model remain complex and nonlinear. The non-linear nature of the equations added with the stochastic uncontrollable input, the wind, makes it difficult to design linear controllers. When the pitch angle θ, the rotor speed Ωro , and the undisturbed wind speed Vun (which is related to rotor speed), are kept constant and the effect of the gravity is neglected, the dynamic equations described in §2-2-1 give a hand to be handled as linear equations. From these set of equations we can derive an Linear Time Invariant (LTI) structure which varies with respect to the variables θ and Vun . Thus, this idea gives an insight that it is possible to model the system as an Linear Parameter Varying (LPV) system (discussed in Chapter 3). Therefore, the challenge is developing an LPV model of the system in which the time varying scheduling parameter are the rotor speed (or wind speed), pitch angle of the rotor blades, and the azimuth angle of the rotor blades. The thesis too, concentrates on the methods of developing an LPV model of wind turbines by accessing the works done so far in this area. In §3-2-1, the work of Bianchi et al. [3] on LPV modeling of wind turbines is discussed, which considers only the rotor rotational dynamics. The work is based on first principle nonlinear equations described in Eq. (2-9) and Eq. (2-12). In Chapter 4 some of the methods used for identifying LPV methods from input-output data is discussed. S.K. Zegeye

Literature survey

2-4 Conclusions and Results

2-4

11

Conclusions and Results

In this chapter it has been discussed that, though the wind energy is free and has the potential to supply the current demand of electricity, it is still expensive. Furthermore, it has been stressed that the input to wind turbines is unpredictable and the first equations governing the dynamics of wind turbines are highly nonlinear, complex, and coupled. The nature of the variation of the equations of the wind turbines with respect to the variation of wind speed and the rotor speed, does show that the system could be considered as time varying linear system. Thus, it is required to develop an LPV model of the system both from the nonlinear equations and input-output data, and evaluate them.

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Wind Turbines

Literature survey

Chapter 3 Linear Parameter Varying Systems

Linear Time Invariant (LTI) models can approximate nonlinear systems in the neighborhood of the linearizing point. The models may not fit well for physical modification and when used out side the operating point . Bamieh and Giarre [2] have proposed that a control system based on gain scheduling is one approach to handle nonlinearities. According to their work Linear Parameter Varying (LPV)1 model can be used to study gain scheduling. The need to model system more accurately results in either non-linear model, which is difficult to use for control purpose, or a set of linearized models along a trajectory which is not restricted to local operating regions [2]. Since non-linear models are not suitable for control, the parametrization of non-linear systems as LPV model can better approximate than LTI model of the systems. Therefore developing a method that can suitably approximate nonlinear systems better than LTI systems is a bottleneck. In this chapter we will first give an overview on LPV systems. Then after, it will be discussed the methods so far used in transforming nonlinear systems to corresponding LPV model. The chapter will end with some concluding remarks.

3-1

Overview on LPV systems

In general LPV systems are described in state space form, where the system matrices are dependent on time-varying parameter. Eq. (3-1) shows an LPV system in state space form. x(t) ˙ = A(p(t))x(t) + B(p(t))u(t) y(t) = C(p(t))x(t) + D(p(t))u(t)

(3-1)

Where x ∈ Rn , u ∈ Rm , y ∈ Rl and the dependence of the system matrix on the parameter p(t) is assumed to be affine. i.e. they can be expressed as: A(p(t)) = Ao +

s X

pi (t)Ai

(3-2)

i

1

The term Linear-Parameter time-Varying was first introduced in Shamma and Athans (1991 )[8]

Literature survey

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Linear Parameter Varying Systems

Where Ao , Ai ∈ Rn×n and i ∈ {1, 2, 3..., s}. All B(p(t)), C(p(t)) and D(p(t)) can be expressed in the same way. Here s is the dimension of the time varying p(t) and i indicates the ith row of the parameter at time t. The system in Eq. (3-1) can be regarded as a special nonlinear system, in which the system non-linearity varies in accordance to the variation of the parameter p(t). i.e. it can be interpreted that at each fixed p(t) the linearization Eq. (3-1) describes the local behavior of the nonlinear plant about the corresponding operating point [18, 27]. If one or more of the parameters are the states of the system, then the system is termed as quasi-LPV [10, 21]. The general description of quasi-LPV systems is expressed as: x˙ A(x1 ) B(x1 ) x = (3-3) C(x1 ) D(x1 ) u y with the states xT = [xT1 xT2 ]. The states x1 are the scheduling parameters whose variations are in the compact set of the scheduling parameters. The parameter p(t) is assumed a prior unknown. However, it can be measured or estimated when the system is operating . As described above LTI models can approximate nonlinear systems in the neighborhood of the linearizing point. As the operating point (linearizing point) of the system varies it fails to model the system. Recently, many researches have been doing researches to design control algorithms for LPV systems [27]. In their statement Verdult et al. [27] stressed that nonlinear systems can be approximated over a whole operating range by taking a weighted combination of local linear models that were obtained in several different operating points. The weighting factors then varies in accordance to the operating point [27]. Some special LPV systems can also be expressed in Linear Fractional Transformation (LFT) form. That is, in some cases it is possible to model or approximate parameter dependent LPV systems as an LFT [18]. The LPV system in Eq. (3-1) can be represented as LFT if all parameter dependent system matrix can be expressed as an LFT form F(M, ∆(p)) for some matrix M , and diagonal p-dependent matrix ∆(p). This can be well explained with the help of Eq. (3-1). The input-output relation of the signals w and z of the block diagram in figure Fig. 3-1 can be expressed as:

Figure 3-1: Fu (M, ∆(p))

Block diagram representation of Upper Linear Fractional Transformation

z = (M22 + M21 ∆(p)(I − M11 ∆(p))−1 M12 )w ∆(p) = diag{p1 In1 , p2 In2 , ..., ps Ins } S.K. Zegeye

(3-4) Literature survey

3-2 LPV modeling of nonlinear systems

15

Where ns are suitable integers and Ins are ns × ns identity matrix.

3-2

LPV modeling of nonlinear systems

It has been described that the scheduling parameter determines the variation of the system dynamics with time. The first step in obtaining an LPV model of nonlinear systems is to identify the parameter on which the system dynamics is dependent. This is not a simple task and has no general formula. It is mostly problem dependent and requires physical understanding of the system understudy . The present theory does not support the reformulation of nonlinear systems into linear/quasi-linear parameter-varying form without, in general, considerable restriction either on the class of nonlinear systems considered or on the allowable operating region . In general any arbitrary nonlinear system can be described by: x˙ = f (x, u) y = g(x, u)

(3-5)

where x ∈ Rn the state , u ∈ Rm the input signal y ∈ Rl the output signal of the system. Certain nonlinear plants can be represented as LPV plants. The nonlinear system is transformed to LPV form by using suitable (nonlinear) transformations [5]. A simple example of particular quasi-LPV formulation of nonlinear system is given in Example 3.1. In the example the matrix A(x) varies with the state x1 . Thus the system described by Eq. (3-7) is called quasi-LPV. In the sequel some of the methods used for obtaining an LPV model from nonlinear systems are discussed.

EXAMPLE 3.1 consider the nonlinear system x˙ 1 = sin(x1 ) + x2 ,

x˙ 2 = x1 + u.

This can be reformulated in quasi-LPV form as: sin(x1 )/x1 x˙ = A(x)x + Bu = 1

3-2-1

1 0 x+ u 0 1

(3-6)

(3-7)

Taylor-Series Expansion Linearization Method

The expression in Eq. (3-1) assumes that the parameter p(t) is an exogenous time varying quantity which takes values in the allowable set . Leith and Leithhead [10] have described that under these conditions, an LPV system is simply a particular form of Linear Time Varying system. They have shown how to obtain an LPV model from nonlinear systems. The text below mainly discusses their work. LPV representation of nonlinear systems are largely associated with series expansion linearization theory and this is reviewed below. The method is also called Jacobian linearization [12, 19].

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Consider nonlinear system described in Eq. (3-5), where f (., .) and g(., .) are differentiable with bounded, Lipschitz continuous derivatives. The set of equilibrium points of the nonlinear system in Eq. (3-5), consists of points, (x0 , u0 ), for which f (x0 , u0 ) = 0

(3-8)

Let Φ : Rn × Rm denote the space consisting of the union of the state, x, and the input u. The set of equilibrium operating points of the nonlinear system in Eq. (3-5), forms a locus of points, (x0 , u0 ), in Φ and the response of the system to a general time-varying input, u, is depicted by the trajectory in Φ. Let also that (˜ x(t), u˜(t), y˜(t)) denote a specific trajectory of the nonlinear system in Eq. (3-5); i.e., x ˜˙ = f (˜ x(t), u ˜(t)),

y˜(t) = g(˜ x(t), u˜(t))

(3-9)

The trajectory, (˜ x(t), u˜(t), y˜(t)), could be an equilibrium point of Eq. (3-5), in which case f (˜ x(t), u ˜(t)) is identically zero and x ˜ is a constant. The nonlinear system in Eq. (3-5), may be reformulated, relative to the trajectory (˜ x(t), u˜(t), y˜(t)), as: ˙ = ∇x f (˜ δx x, u ˜)δx + ∇u f (˜ x, u ˜)δu + εf

(3-10)

δu = u − u ˜, δy = y − y˜, δx = x − x ˜

(3-12)

δy = ∇x g(˜ x, u ˜)δx + ∇u g(˜ x, u ˜)δu + εg

(3-11)

where, x, u ˜) − ∇x f (˜ x, u ˜)δx − ∇u f (˜ x, u ˜)δu εf = f (x, u) − f (˜ εg = g(x, u) − g(˜ x, u ˜) − ∇x g(˜ x, u˜)δx − ∇u g(˜ x, u ˜)δu

(3-13) (3-14)

From Taylor series expansion theory |εf | ≤ σ(|δx| + |δu|)2 , |εg | ≤ σ(|δx| + |δu|)2

(3-15)

where σ is a finite positive constant. Hence, provided |δx| and |δu| are sufficiently small, the dynamics in Eq. (3-10) and Eq. (3-11), can be approximated by the linear time-varying system δx ˆ˙ = ∇x f (˜ x(t), u ˜(t))δˆ x + ∇u f (˜ x, u ˜)δu δyˆ = ∇x g(˜ x(t), u˜(t))δˆ x + ∇u g(˜ x, u ˜)δu

(3-16) (3-17)

The system described by Eq. (3-16) and Eq. (3-17), is in LPV form (the parameter is an exogenous time-varying quantity independent of the state) and is simply the first order Taylor series expansion of the nonlinear system in Eq. (3-5), relative to the trajectory, (˜ x(t), u˜(t), y˜(t)). The system described by Eqs. (3-16) and (3-17), approximates Eqs. (3-10) and (3-11), in the sense that the solution to Eqs. (3-16) and (3-17) approximates the solution to Eqs. (3-10) and (3-11). Moreover, the system described by Eqs. (3-10) and (3-11) is exponentially stable, locally to the trajectory (˜ x(t), u˜(t), y˜(t)), if and only if Eqs. (3-16) and (3-17) is stable. Since the transformations in Eq. (3-12) is confined to specific trajectory, the transformations in Eq. (3-12), are static.

S.K. Zegeye

Literature survey


17

Owing to the requirement that |δx| and |δu| are sufficiently small, the series expansion linearization in Eqs. (3-16)) and (3-17), of the nonlinear system in Eq. (3-5), is only valid within a small neighborhood about (˜ x(t), u ˜(t), y˜(t)). The limitation is inherent to the series expansion linearization. Since it is a first order approximation, it could lead to divergent behavior (with respect to the nonlinear system) for large control inputs [12]. It has been suggested in Marcos and Balas [12], that the local nature of the system can be improved by using higher order terms in Taylor-series expansion, but this could lead to impractical implementation. It has to be noted that conceptually LPV model is different from a family of linearized systems. This is because the latter is a collection of dynamical systems defined by perturbations and the former is a single dynamic system. In general it is impossible to capture the transient behavior of the nonlinear system by the method. However, for certain systems it is possible to account for its essential features [12]. The advantages gained in using Taylor-series linearization (Jacobian linearization) are:1) it is a well known method, 2) applicable for wide class of nonlinear systems with out restriction of affine dependence of input u, and 3) suitable to analyze the local stability of nonlinear systems which is matched to the stability of quasi-LPV system [19].

LPV model of wind turbines using Taylor series method Here the work of Bianchi et al. [3] on LPV modeling of wind turbine is discussed. In their work, Bianchi et al. [3], have considered Eqs. (2-9) and (2-12). For a matter of brevity the equations are repeated below (Note that the symbols used are modified to keep consistency in the survey). They have not included the effect of the tower, the flap angle of the blades and effect of gravity. Jro Ω˙ ro = Tro (t) − Q Q˙ = kro ε˙ + dro ε¨ro Jge Ω˙ ge = Q − Tge

(3-18) (3-19) (3-20)

Where Q is shaft torque, and Tro , Tge are defined in Eqs. (2-9) and (2-13). Hence, the given set of equations can be expressed as LPV system using Taylor series method, when the torque of the generator is expressed as linear equation (see: Eq. (3-21)) under the assumption that the slip is small and when the torque coefficient is approximated as second-order polynomial (see: Eq. (3-22)) [4]. Tge = Tge,1 Ωge + Tge,2 ωsl 2

Cq = c2 λ + c1 λ + c0

(3-21) (3-22)

Where ωsl is single dominant resonance frequency and Tge,1 and Tge,2 are linearizing constants. Then, linearizing Tro (t) = 12 πρR3 Cq (λ)υ 2 , at the generic operating point defined by the mean wind speed υm and the mean rotor speed Ωro,m , the aerodynamic torque is ˆ ro + b(υm , Ωro,m )ˆ Tro = a(υm , Ωro,m )Ω υ Where

Literature survey

ˆ ro = Ωro − Ωro,m , Ω

(3-23)

υˆ = υ − υm S.K. Zegeye

18


a(υm , Ωro,m ) =

∂Tro πρR4 c1 = υm + πρR5 c2 Ωro,m ∂Ωro υm ,Ωro,m 2

∂Tro πρR4 c1 b(υm , Ωro,m ) = = Ωro,m + πρR5 c0 υm ∂υ υm ,Ωro,m 2

Finally on substitution of Eq. (3-23) to Eq. (3-18), gives the LPV model of the Wind Energy Conversion System (WECS) as: x˙ = A(p)x + B(p)u, where

y = C(p)x

(3-24)

ˆ ro Q Ωge ], uT = [ˆ xT = [Ω υ ωsl ], pT = [υm Ωro,m ]

A=



a(υm ,Ωro,m ) Jro   dro a(υm ,Ωro,m ) kro + Jro



0

1 − Jro −dro J1ro +



b(υm , Ωro,m ) d a(υm ,Ωro,m ) B =  ro Jro 0

1 Jge

0



Tge,2  Jge  , Tge,1 − Jge

dro

1 Jge

0 dro

Tge,1 Jge

−



  − kro  

Tge,1 Jge

0 1 0 C= 0 0 1

The resulting equation in Eq. (3-24) gives the LPV model of the wind turbine for the rotational dynamics.

3-2-2

Function Substitution Method

The method, function substitution, was first proposed in Tan [20] with nonlinearity in control input. Note that quasi-LPV systems must be linear with respect to non-scheduling states and control input. In the paper of Marcos and Balas [12] it is proposed that the substitution of decomposition function (Fdec ) by scheduling parameter-dependent functions linear in the scheduling vector. The decomposition function is obtained by combining all the terms of the nonlinear system that are not both, affine with respect to the non-scheduling states and control input, and function of the scheduling vector alone. The decomposition is formed after the coordinate is changed with respect to a single selected equilibrium point [12]. The method is used to formulate a quasi-LPV model over an entire operating region of the nonlinear system, including non-trim points [19]. In order to perform the substitution let choose an equilibrium point (x1o , x2o , uo ) where x1 are scheduling states and x2 are non-scheduling states, while the subscript ’o’ indicates the states are the equilibrium points. Performing coordinate transformation we get: x ˜1 = x1 − x1o , x ˜2 = x2 − x2o , u ˜ = u − uo

(3-25)

Let assume the nonlinear equation in Eq. (3-5) can be represented as: x˙ = f1 (x1 )x + f2 (x1 )u + f3 (x1 ) S.K. Zegeye



19

where f1 , f2 and f3 are continuous mapping function : Rnx1 7−→ Rnx ×nx , Rnx1 7−→ Rnx ×nx , and Rnx1 7−→ Rxn respectively. Then using Eq. (3-25) we can write the nonlinear system in Eq. (3-26) as: A11 (x1 ) A12 (x1 ) x ˜1 B1 (x1 ) x ˜˙ 1 = + u ˜ + Fdec (x1 ) (3-27) ˜2 A21 (x1 ) A22 (x1 ) x B2 (x1 ) x ˜˙ 2 Where the decomposition function (Fdec (x1 )) is given by: A11 (x1 ) A12 (x1 ) x1o B1 (x1 ) Fdec (x1 ) = u + f3 (x1 ) + A21 (x1 ) A22 (x1 ) x2o B2 (x1 ) o

(3-28)

The main idea of this method is to reformulate the decomposition function in Eq. (3-28) into quasi-LPV function [19] form such as:   e11 (x1 ) e12 (x1 ) . . . e1nx1 (x1 )   .. .. .. ˜1 Fdec (x1 ) = E(x1 )˜ x1 =  ... (3-29) x . . . en1 (x1 ) en2 (x1 ) . . . ennx1 (x1 )

Where n is the order of the system, nx1 is the length of scheduling state vector, and the matrix E is unknown matrix to be determined and eij is the ij element of the matrix E. In Marcos and Balas [12], it is has been shown that the decomposition functions in Eq. (3-29) is exact if the columns of the matrix E, which are given by e.j (x1 ) are selected as: e.j (x1 ) =

Fdec (x1 , x1o , x2o , uo )˜ x1j x ˜211 + x ˜212 + . . . + x ˜21nx

(3-30)

1

It is also noted in the same paper that these decomposition functions have to be smooth and well-defined to avoid controller synthesis problems arising from discontinuities in the quasiLPV model. It is necessary to select grid points of the scheduling parameters in order to make the problem finite dimensional [12]. Once the grid points are selected the particular solution of Eq. (3-30) is obtained easily [12]. Since the problem is an under-determined problem, Eq. (3-29) has infinite number of solutions [19]. Hence, numerical approximation is used at the grid points, which could lead to approximation errors [12, and the reference therein]. Hence, two linear programming are used to smooth (minimize the variation of each matrix element of E) the decomposition functions [12, 19]. The final quasi-LPV form of the nonlinear system is then written as: x ˜˙ 1 x ˜1 B1 (x1 ) = Af + u ˜ (3-31) x ˜2 B2 (x1 ) x ˜˙ 2 where A11 (x1 ) A12 (x1 ) Af = + [E|0nx ×nx2 ] A21 (x1 ) A22 (x1 )

(3-32)

The nice feature of this method is that the solution of the quasi-LPV system is close to the nonlinear system. This is because the equality constraint in Eq. (3-29) is satisfied at the selected grid points [19].

Literature survey

S.K. Zegeye

20


However this method has disadvantages such as: strong dependence of the selected reference(equilibrium) point on the result of the quasi-LPV model and possible misrepresentation of local stability of the original nonlinear system over the locus of equilibrium points [19]. Shin [19] has also described that the matrix Af in the quasi-LPV model can be different when different equilibrium points are selected as a reference. This limitation arising from the selection of the correct reference point is relaxed in the work of Shin [19]. He also suggested a new function substitution method. In his method, Shin [19], the quasi-LPV model of the nonlinear system describes the nonlinear over the entire operating region. The model can represent local stability of the nonlinear system. In this method, many reference points can be selected along the trajectory of the locus of the equilibrium points to preserve the stability of the system. The text below follows his work. Let us adopt the definition of existence of equilibrium point in an operating region, which we will use it in the sequel.

Defnition 3-2.1 Suppose there exists the set of E of pre-calculated equilibrium points. Given any point x¯1 in the operating envelope P, there exists an equilibrium point x ˜1e such that x ˜1e = arg min kW [¯ x1 − x ˜1e ]k , x ˜1 ∈ P (3-33) x ˜1e ∈E

Where W is a weighting matrix to compensate physical unit difference of each state.

The details of the methodology developed by Shin [19], as described in his report, is as follows: First, grid points are generated over the entire operating envelope P and each grid point is assigned its own reference point to reformulate the nonlinear function Fdec (x1 ) in Eq. (3-28) into quasi-LPV form. A reference point for each grid point is developed using Eq. (3-33). Second, the matrix E in Eq. (3-29) is written in terms of other variables φi to satisfy the equality constraint. When a grid point x ¯1 is not equal to reference point x1r , Eq. (3-29) is written as:       Fdec1 (¯ x1 ) e1. [¯ x1 − x1r ]T eT1.    ..    .. .. x1 − x1r ] =  (3-34)  =  .  [¯   . . Fdecn (¯ x1 )

en.

[¯ x1 − x1r ]T eTn.

The possible solutions of eTi. in Eq. (3-34) can be written as: eTi. = eTi.p + N [¯ x1 − x1r ]T φi

(3-35)

Where the particular solution eTi.p is

eTi.p = [¯ x1 − x1r ]([¯ x1 − x1r ]T [¯ x1 − x1r ])−1 Fdeci ,

(3-36)

N [¯ x1 − x1r ]T is the null space of [¯ x1 − x1r ]T , and φi is unknown vector associated with the null space dimension. Using Eq. (3-35), the matrix Af is written as: Af = Ap + [ΦT N ([¯ x1 − x1r ]T )|0n×nx2 ] S.K. Zegeye



21

where A11 A12 Ap = + [Ep |0n×nx2 ], A21 A22

(3-38)

the matrix Ep is the collection of eTi.P and the matrix Φ is the collection of φi. , which is determined by Ruth-Harwitz for small order systems or by solving the Bilinear Matrix Inequality (BMI) given by: ATf K + KAf < 0, K > 0, and x1 ∈ P

(3-39)

where P is the parameter set. When a grid point is equal to a reference point x1r , it is easily noticed that (Fdec (x1 )) and x ˜1 are zero. The particular solution of Eq. (3-36) can not be defined since it is a singular point. For this case, the matrix E defined in Eq. (3-29) is defined as: E = lim

x ¯1 →x1r

Fdec (¯ x1 ) x ¯1 − x1r

(3-40)

The matrix Af is calculated using Eqs. (3-32) and (3-40) for a quasi-LPV model at the grid point. The matrix obtained is exactly the same as the matrix obtained by the Jacobian linearization at the grid point.

3-2-3

State Transformation Method

This method is based on the transformation of the nonlinearity of the system in terms of the non-scheduling states and control input into a function of the scheduling states. This method was first introduced in Carter and Shamma [5]. The function selected has to be differentiable. Marcos and Balas [12] also indicated that in order to undergo the transformation of the nonlinear system into quasi-LPV form (whose state space are independent of the non-scheduling states and control input), the size of the scheduling states and the control input has to be the same. As for all LPV systems the scheduling variables are assumed to be available for measurement. To have good insight to the method let us consider general state equation of quasi-LPV model given in Eq. (3-27). Let us assume there exists a set of continuous differentiable functions of the scheduling states x2eq (x1 ) and ueq (x1 ) such that for every of the scheduling variable the system is in steady state [12]: 0 A11 (x1 ) A12 (x1 ) x1 B1 (x1 ) F (x ) = + ueq (x1 ) + dec1 1 0 A21 (x1 ) A22 (x1 ) x2eq (x1 ) B2 (x1 ) Fdec2 (x1 )

(3-41)

On subtraction of Eq. (3-41) from Eq. (3-27) it is obtained that Literature survey

x˙ 1 0 A12 (x1 ) x1 B1 (x1 ) = + (u − ueq ) x˙ 2 0 A22 (x1 ) x2 − x2eq B2 (x1 )

(3-42) S.K. Zegeye

22


On substitution of x˙ 1 into the time derivative of the trim function x2eq (x1 ), we can arrive at the following quasi-LPV model of the output nonlinear system. # " 0 A12 (x1 ) x˙ 1 x1 = ∂x x˙ 2 − x˙ 2eq 0 A22 (x1 ) − ∂x2eq |x1 A12 (x1 ) x2 − x2eq 1 " # B1 (x1 ) (u − ueq ) (3-43) + ∂x B2 (x1 ) − ∂x2eq |x1 B1 (x1 ) 1 As can be observed from the final quasi-LPV model in Eq. (3-43), the decomposition function disappears. The quasi-LPV equation represents the nonlinear system through an exact transformation. The main problem in this method is the existence of the trim function or their validity within the feasible region of the real nonlinear system [12]. Hence, it is only possible to insure that the model obtained is valid in the restricted envelop. As discussed above there are variety of methods which can be used in developing LPV models of nonlinear system. There are also different methods (such as velocity method which can be categorized under Jacobian linearization , higher-order series expansion, mean value theorem methods and the like). These method are extensively discussed in Leith and Leithhead [10]. However, every method can not be applied to any problem with out limitation. All the transformation methods are problem dependent, hence insight to the problem is fundamental.

3-3

Conclusions and Results

In this chapter we discussed LPV systems and their advantages relative to single LTI systems. It is also described that an LPV model can better describe the dynamics of a nonlinear system. Other than Linear Time Varying (LTV) systems in LPV systems the scheduling parameter is not know a prior, but available during the operation of the system. It was also discussed that the selection of scheduling variables is not a simple task, which is the crucial step in an LPV formulation of nonlinear systems. Furthermore, a review has been given on the methods that can be used to formulate quasi-Linear/Linear Parameter Varying model of nonlinear system. We also have discussed a work on LPV formulation of wind turbines. The LPV model developed for wind turbines were not considering the effect of the tower, gravity, flap, and pitch angle of blades. This is an indication that further work is required. It has been also pointed out that the methods for LPV formulation, are not globally applicable to any nonlinear system.

S.K. Zegeye

Literature survey

Chapter 4 Identification of LPV systems

In the previous chapter we showed how to obtain an Linear Parameter Varying (LPV) model of a system from first principle modeling. However, all the methods described show how to formulate an LPV model from existing nonlinear equation of the system. Thus, in formulating LPV models using the methods discussed, we have to have first principle models of the nonlinear system. Nevertheless, developing first principle models is mostly difficult, if done, the models are too complex for huge plants. However, first principle models can be used as a bases to give some insight to the process. Experimental modeling is basically called System Identification, where measurement of several variables of the process are taken and a model is developed to match the measured data as good as possible [22]. In the first section of the chapter system identification will be reviewed. While in the second section subspace identification as applied to Linear Time Invariant (LTI) systems and LPV systems will be discussed, giving much emphasis on LPV systems. The chapter will end with conclusion and results.

4-1

Overview of System Identification

System identification is nothing but the development of a model of a system based on measured variables. In system identification, the model is obtained as to give the optimal solution which reduces model mismatch. Hence, system identification models consider the model mismatch due to some unknown nonlinearity that can be difficult to describe using first principle. In general, in all system identification methods, experiment design such as determining the sampling time and type of input are fundamental steps. These steps are not discussed in this survey. It is assumed that the reader has considerable knowledge on the subject. It is also crucial determining the identification criterion to guarantee some degree of certainty in the model match. The input-output description of an LTI data generating finite dimension Literature survey

S.K. Zegeye

24

Identification of LPV systems

system can be described as in Eq. (4-1). y(t) + a1 y(t − 1) + · · · + ana y(t − na ) = b1 u(t − 1) + · · · + bnb u(t − nb ) + ε(t)

(4-1)

with the data set {y(t), u(t)}t=1,...,N , and where the parameter vector θ ∈ R2(n−1) is composed as θ = [a1 , . . . , ana , b1 , . . . , bnb ]T and ε is the error, modeling the model mismatch and the noise in the data generating system [22]. Hence a model set is obtained by varying the vector θ over a parameter set Θ ⊂ R2(n−1) . This shows that in the input-output description determining the model structure (number of parameters/order of system) is a step before starting the identification of the parameters. In such a method the identification criterion is developed on the bases of the error ε in different ways. For example the most popular is the least square criterion. VN (θ) =

N 1 X 2 ε (t, θ) N t=1

(4-2)

and a parameter estimate is constructed by finding the value of θ that minimizes VN (θ) [22]. This can be written as: ! θˆN = arg min VN (θ) (4-3) θ∈Θ

meaning that θˆN is the value of θ that minimizes the sum-of-squares criterion. Note that there are also different identification methods, which use different approaches to define the identification criterion. It is also possible to determine LTI model of a system by structuring the input output data into block-Hankle matrices without the need of parametrization. This method is known as subspace identification. In general there are many identification methods developed for LTI systems. Verdult et al. [26] have described that there exists multitude identification methods for Multi Input Multi Output (MIMO) LTI systems but the choices are limited when dealing with MIMO nonlinear systems. They point out also that in dealing with MIMO systems state space models offer better advantage than input-output description. Verdult and Verhaegen [24] and the reference there in, confirm that subspace identification is well accepted method for identifying MIMO linear systems. Verdult et al. [26] point out that subspace identification methods are numerically attractive. This is because the method does not require some particular parametrization. The method does not require determination of system order before undergoing the identification. In the following section of this chapter subspace identification is extensively discussed.

4-2

Subspace Identification

Subspace identification algorithms are based on concepts from system theory, linear algebra, and statistics. This method directly determines an LTI state space model of a system from S.K. Zegeye

Literature survey

4-2 Subspace Identification

25

the given input output data. For a data generating LPV/LTI system described by xk+1 = Ak xk + Bk uk + wk yk = Ck xk + Dk uk + vk

(4-4)

with E 1 [(wpT

vpT )T (wqT

vqT )]

Q = T S

S δ ≥0 R pq

(4-5)

Where the input vector uk ∈ Rm and the output vector yk ∈ Rl are the observation at a time instant k of m inputs and l outputs respectively. The vector xk ∈ Rn is the state vector of the discrete system, and wk and vk are process and measurement noise respectively. The subspace identification problem is then, determination of the set of matrices Ak , Bk , Ck , Dk , Q, R, S, and the order of the system n for a given data set (input and output). The matrices are constant for every k when dealing with LTI system, while they are assumed to be affine dependent on measurable scheduling parameter pk in the case of LPV systems. In the identification method stated above, the matrix pair (Ak , Ck ) is assumed to be observable, which implies that all modes in the system can be observed in the output data 1 yk and thus can be identified [6, 28]. The matrix pair (Ak , [Bk Q 2 ]) is also assumed to be controllable, in turn implies that all the modes of the system can be excited by either the deterministic input uk and/or the stochastic input wk [6, 28]. The subspace identification method is based on the fact that by sorting the input and output data into structured block-Hankel matrices, it is possible to retrieve column space of observability matrices. In the next two subsections below, some literature in subspace identification as applied to LTI systems and LPV systems will be reviewed. Generally the concept discussed above applies as a principle to both systems. The differences as applied to these systems will also be reviewed. The current challenge of subspace identification in LPV systems will be addressed too in the second subsection.

4-2-1

LTI Subspace Identification

As it has been described above the first step in subspace identification is determining the state sequence of the system. An important achievement of subspace identification is, its capability of determining the Kalman filter states directly from the input-output data without knowing the mathematical model of the system using linear algebra (Singular Value Decomposition and QR factorization) [6]. Any LTI systems with white measurement and process noise can be expressed in innovation form as given below. xk+1 = Axk + Buk + Kek yk = Cxk + Duk + ek Literature survey

(4-6) S.K. Zegeye

26


with ek is white noise sequence and K is Kalman gain [28]. The text below follows the text of Verhaegen et al. [28] on Filtering and System Identification: An introduction to using Matlab Software. The state of the system described by Eq. (4-6) with initial state x0 is given by k

xk = A x0 +

k−1 X

Ak−i−1 Bui

(4-7)

i=1

Hence by invoking the output equation of Eq. (4-6) the relationship between the input data s−1 batch {uk }s−1 k=0 and the output data batch {yk }k=0 can be expresses as        C u0 y0 D 0 0 ··· 0  y1   CA   CB   D 0 ··· 0        u1     y2   CA2   CAB CB D · · · 0   u2  =  x0 +     ..   ..   .. .. .. ..   ..  ..  .   .     . . . . . .  s−1 s−2 s−3 s−4 ys−1 CA CA B CA B CA B · · · D us−1 | {z } {z } | Os

Ts



Il CK CAK .. .

0 Il CK .. .

··· ··· ··· .. .

0 0 Il .. .

 0  0   0  ..   . 

e0 e1 e2 .. .



      +      s−2 s−3 s−4 CA K CA K CA K · · · Il es−1 | {z } Ss

(4-8)

where s is any positive integer, greater or equal to the order of the LTI system. The matrix Os is referred as the extended observability matrix. The same relation can be developed by shifting the data batch by a unit step to obtain the data equation:   y0 y1 · · · yN −s  y1 y2 · · · yN −s+1     .. .. . . ..  = Os [x0 x1 · · · xN −s ] | {z }  . . . .  

  +Ts   |

u0 u1 .. . us−1

|

ys−1 ys · · · {z

Y0,s,N

X0,N

yN

}

  u1 · · · uN −s e0 e1   u2 · · · uN −s+1   e1 e2 .. . . ..  +Ss  .. ..  . . . .  . us · · · uN es−1 es {z } | U0,s,N

 · · · eN −s · · · eN −s+1   ..  .. . .  ··· eN {z }

(4-9)

E0,s,N

Where in general s n) Un (1 : (s − 1)l, :)AT = Un (l + 1 :, sl, :)

(4-18)

While the matrices BT and DT , along with the initial state xT (0) = T −1 x(0) is obtained by solving a least squares problem " ! #  x (0)  k−1 T X −1 T  vec(B y(k) = CˆT AˆkT u(τ ) ⊗ CˆT Aˆk−τ (u(k) ⊗ I ) (4-19) T ) l T τ =0 T) {z } | vec(D | {z } φ(k)

θ

where AˆT and CˆT are the estimates of AT and CT which are obtained from Eq. (4-17). Hence the under-braced vector θ can be obtained by solving the optimization problem N −1 1 X min ky(k) − φ(k)T θk22 θ N

(4-20)

k=0

4-2-2

LPV Subspace Identification

So far in the above section, it has been discussed about subspace identification for an LTI systems. The advantages of subspace identification over classical identification methods were also addressed. Due to the nice structure of LTI systems in Eq. (4-8), it has been shown that the system matrices can be estimated by solving least square problems. Different methods have been proposed for the identification of LPV systems. Lee and Poolla [9] has formulated an output-error identification problem for LPV systems, whose parameter dependence can be written as a Linear Fractional Transformation (LFT). They presented nonlinear programming to estimate parameters, by minimizing a prediction error-based cost function. In the work of Previdi and Lovera [17], it is shown that LPV models which can be expressed as LFT can be identified with hybrid linear/nonlinear procedure. Bamieh and Giarre [1] have also presented an identification method for input-output parameterized model by reducing to linear regression and have provided compact formula for the corresponding least mean square and recursive least square algorithms. In Verdult et al. [26], identification of fully parametrization state space systems by gradient search method is implemented. In the paper output error identification was implemented to identify the fully parameterized system matrices. Since the cost function defined is nonlinear and nonconvex, Verdult et al. [26] proposed gradient search method. To gain the advantage of subspace identification, at a glance to the structure of LPV systems, one can propose similar procedure to identify LPV systems as for LTI systems. However, LPV systems do not have the nice structure of the extended observability matrix as LTI systems. One can think, to estimate the states of the system, based on the work of Overschee and Moor [15] and Picci and Katayama [16] for LTI systems. On application of the algorithm to an LPV systems, it creates two fundamental problems, arising from the dimension of the S.K. Zegeye

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29

data matrices. Verdult and Verhaegen [24] have shown with an example that the size of the data matrix increases exponential with the order of the system. They also addressed the two problems as: increase of the rows of data matrix and requirement of enormous amount of samples to have larger number of columns than rows in the data matrix. In their paper they provide a solution which can give an approximate of the estimates by considering the dominant rows of the data matrix. For a matter of clarity, the problem of LPV subspace identification is discussed below.

LPV subspace identification problem/solution In most literature, the LPV system considered is a system described by Eq. (4-21) and Eq. (4-22), where in which the state and input matrices are affine dependent on the scheduling parameter pk . Felici et al. [7] have also considered periodically varying parameter which will be discussed in next subsection. Here below, an LPV system with a scheduling parameter, varying nonlinearly is considered. The LPV system can be written as: uk xk +B + wk (4-21) xk+1 = A pk ⊗ xk pk ⊗ uk

xk uk yk = C +D + vk pk ⊗ xk pk ⊗ uk

(4-22)

where A = [A0 , A1 , A2 , ..., Ar ] ∈ Rn×nr , B = [B0 , B1 , B2 , ..., Br ] ∈ Rn×mr , C = [C0 , C1 , C2 , ..., Cr ] ∈ Rl×nr , D = [D0 , D1 , D2 , ..., Dr ] ∈ Rl×mr , ⊗ denotes Kroncker product [27], xk ∈ Rn represents the unknown state, uk ∈ Rm is the input, yk ∈ Rl is the output, and pk ∈ Rr the time-varying parameter vector. Letting Xj,N +j−1 = [xj xj+1 xj+2 . . . xj+N −1 ] and similarly for Uj,N +j−1 , Yj,N +j−1, Pj,N +j−1 , Wj,N +j−1, and Vj,N +j−1 , we can rewrite the system in (4-21) and (4-22) more compactly as: Xj+1,N +j = [A B]Φj,N +j−1 + Wj,N +j−1

(4-23)

Yj,N +j−1 = [C D]Φj,N +j−1 + Vj,N +j−1

(4-24)

Where

Φj,N +j−1



 Xj,N +j−1 Pj,N +j−1 ⊙ Xj,N +j−1   :=    Uj,N +j−1 Pj,N +j−1 ⊙ Uj,N +j−1

(4-25)

and the symbol ⊙ represents the Khatri-Rao product [27], which is column wise Kronecker product. Thus the problem of identification for the above system can be stated as: given an LPV (special structure) data generating system by Eqs. (4-23) and (4-24), finite measurement input uk , output yk , and time varying parameter pk , determine the system matrices A, B, C, and D up to a similarity transformation. Principally the method described in §4-2-1 can be used to identify the system matrices. When the state sequence is first estimated, the system Literature survey

S.K. Zegeye

30


matrices can be estimated by solving two least squares problems based on Eq. (4-23) and Eq. (4-24). The estimated system matrices are computed as: ˆ B] ˆ =X ˆ 2,N +1 ΦT (Φ1,N ΦT )−1 [A, 1,N 1,N ˆ D] ˆ = Y1,N ΦT (Φ1,N ΦT )−1 [C, 1,N

(4-26)

1,N

ˆ B, ˆ C, ˆ D, ˆ and X ˆ 2,N +1 are the estimates of A, B, C, D, and X2,N +1 respectively. where A,

However, for practical application, the solution in Eq. (4-26) is not suitable for the large data matrix and requirement of enormous amount of data samples in determining the state sequence estimates of the system [23]. An extensive and more elaborative explanation is given in the PhD thesis of Verdult [23], to estimate the state sequence of LPV system. The main drawback of the method is that the system matrices are obtained from approximate of the states. Verdult and Verhaegen [24] have shown a method to overcome the problem in the dimension of the data in estimating the states of the system. In their work, Verdult and Verhaegen [24], have solved the dimensionality problem by using only the most dominant rows of the data matrix. The two main disadvantages of the proposed method are 1) The identified LPV model is obtained from subsets of the rows of the data, which might loss relevant information from the removed row data. 2) In order to select the dominant rows long computation time is needed. Hence it is worthwhile to refine the model using nonlinear optimization.

The expensiveness in computation of the method proposed by Verdult and Verhaegen [24], is solved in their paper [25]. They applied a Kernel method to handle the dimensionality problem. However, the method introduces large variance error, which they optimized it by allowing some bias. In order to balance the bias and the variance they have integrated regularization methods. Owing to the huge nature of the matrix equations describing the above two methods of Verdult and Verhaegen [24, 25], it has been only explained verbally. An interested reader can get thorough information in the aforementioned papers.

LPV identification method for periodic systems For periodic systems the weighted combination of the system matrices given in Eq. (4-21) repeat periodically as a consequence of the periodicity of the time varying parameter pk . In practice there exists a lot of systems which are periodic in nature such as: CD/DVD players, helicopter rotor, wind turbines and the like. However, less work has been done for the identification of rotating (periodic) dynamics [27]. In the sequel some of the works on periodic LPV systems is discussed.

Verdult et al. [27], have shown a way to identify periodic systems with a structure given S.K. Zegeye

Literature survey


31

below.

xk+1 =

N X

A0 +

n=1 N X

+ B0 + yk =

(Acn cos(nΩk) + Asn sin(nΩk)) xk

C0 +

+ D0 +

(Bcn cos(nΩk) + Bsn sin(nΩk)) uk

n=1 N X

(4-27)

(Ccn cos(nΩk) + Csn sin(nΩk)) xk

n=1 N X

(Dcn cos(nΩk) + Dsn sin(nΩk)) uk

(4-28)

n=1

It can be easily seen that this model class is specific case of an LPV system. The time varying parameter can be written as: pk = [cos(Ωk) sin(Ωk) cos(2Ωk) sin(2Ωk) . . . cos(N Ωk) sin(N Ωk)]T

(4-29)

In their paper, Verdult et al. [27], have used nonlinear optimization method based on projected gradient search to obtain fully parameterized solution of the LPV system. The cost function is defined as shown in Eq. (4-30). VN (θ) :=

N X k=1

kyk − yˆk (θ)k22

(4-30)

with yk denoting measured output data, yˆk (θ) denoting the output of the estimated LPV model and the model parameter vector θ begin given by: A0 Ac1 As1 . . . AcN AsN B0 Bc1 Bs1 . . . BcN BsN θ = vec (4-31) C0 Cc1 Cs1 . . . CcN CsN D0 Dc1 Ds1 . . . DcN DsN The method discussed is based on nonlinear optimization and, assumes the periodicity of the time varying parameter pk as the structure given in Eq. (4-29). Felici et al. [7] have shown subspace identification method to handle LPV systems with periodically varying scheduling vector pk . The method applied follows a similar way as is used in LTI subspace identification. However, since the extended observability matrix is periodic, they decomposed it into a product of known matrix Mk and unknown matrix Sk . That is Mk composes of elements derived from the periodic scheduling parameter pk and Sk composes of matrix derived from A and C of the LPV system. The main attainment in the paper is in coming with idea of the decomposition and transformation of the different observability matrices into common bases. The method comes with three main advantages, namely: 1) The method does not use nonlinear optimization. 2) The method do not apply approximation of states to estimate the system matrices. 3) The periodicity of the scheduling variable is only required during the course of identification. The third point is an excellent attainment of the work done. That is, the method can be used for the identification of non-periodic LPV system only by running the system periodically during course of identification. As in the case of the LPV subspace identification, the demerit in the method proposed is, large dimension of the data matrix Mk . In similar fashion as Verdult and Verhaegen [25] they solved the problem by using Kernel method. For a detailed discussion, see the reference [7]. Literature survey

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32

4-3


Conclusions and results

So far in this chapter, it has been described the algorithm behind subspace identification using LTI systems, and some of the works that has been done with respect to the identification of LPV systems. we also gave an over view of classical and subspace identification in the subject of LPV systems. As it was described in the subspace identification of LPV systems, there appears to exist a lot of challenges in the dimension of the data matrices, and the assumption of the scheduling parameter dependence of the system matrices. At last but not least, it is reviewed on literatures which deal with special (periodic) type of LPV systems. The method could be applied to wind turbines, which needs further investigation. It has been shown that, the method applied for periodic systems still assumes affine dependence of the system matrices. In the methods discussed one can ask what if the dependence is assumed nonlinear (unknown)?, which is open area for research.

S.K. Zegeye

Literature survey

Chapter 5 Model Validation

In the last two chapters of the survey, it was only discussed on the techniques so far used in developing Linear Parameter Varying (LPV) model of a system from first principle nonlinear models, and from data. However, validation techniques of models developed was left open. As most people agree, naturally people do evaluate the performance of systems relatively. There is no absolute reference upon which the perception of accuracy, speed of response, and the like can be compared. Although the level of accuracy of a model describing a system is dependent on the application of the system, it is worthwhile to have a general scheme to validate a model. This chapter discusses the techniques used to validate (precisely speaking invalidating) of a model mainly based on the work of Mazzaro and Sznaier [13].

5-1

Overview on Model Validation

Both the methods discussed in Chapters 3 and 4 in developing LPV model of a system, only give the nominal model of the system understudy. However, for a control engineer a complete description of the system model including the nature of the uncertainty is invaluable. LPV Model development too, let it be from nonlinear equations of the system or from real time data, has to be accompanied with the specification of the uncertainty incorporated in the model. Model (in)validation can be used in determining the uncertainty that the nominal model incorporates for the data considered. In validating any system model, one needs either real time data of the system or desired characteristic of the model to be developed. In any of the cases, the model developed is compared with some reference. As has been discussed in Chapters 3 and 4, It is possible to get an LPV model of a system either form nonlinear model of the system or from real time data of the system. The loss in certainty of the LPV model obtained from nonlinear system, includes the errors which account to the assumptions made in developing the nonlinear model. Literature survey

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34

Model Validation

Where as the uncertainty for a model developed from identification techniques only results from the limitation of the technique used. Hence while creating comparison between the uncertainties of the two models, it is worthwhile to take this point into account.

Figure 5-1: General scheme in model development and (in)validation

Fig. 5-1 shows the scheme that is used while comparing the models developed from nonlinear model and data. As the figure describes both the models are validated with real time data. In order to avoid coincidence of the response of the model developed with identification techniques the validation data has to be different from the identification data [28]. The figure also shows, after (in)validation of the LPV models obtained using the two methods, different uncertainties. This is to show that the resulting uncertainty is different for most real systems. Most Identification methods use Variance Accounted For (VAF) as means to assess the quality of a model. It is defined as [28]: V AF (yk , yˆk ) =

1−

1 N

PN

ˆk k22 k=1 kyk − y 1 PN 2 k=1 kyk k2 N

× 100%

(5-1)

Where yk is the real time output, yˆk is the prediction output and N is the size of the data. The higher the VAF is the better is the model (lower prediction error). While validating a model most frequently auto and cross correlation of the residual are used. These methods give the certainty level of the model with respect the real time data. However, it does not specify the structure of the uncertainty in the system. Hence, the resulting model is not suitable for control algorithms which consider uncertainty into account (e.g. Robust control). Now let consider the setup shown in Fig. 5-2, which describes uncertain system M∆ as the lower fractional representation of known model MP and unknown bounded uncertainty k∆k∞ ≤ δ < 1 (i.e. Mδ = Fl (MP , ∆)). Where the subscript P is a diagonal matrix of the time varying parameters defined as P = diag{Ip1 , Ip2 , ..., Ipn }, with n is the number of the parameters. The system described has a sequence of input data u and noise, ω, corrupted S.K. Zegeye

Literature survey

5-2 LMI approach to LPV model (in)validation

35

output data y. The a prior noise considered is of the form : N

X . ω = [ω0 , ω1 , ..., ωN −1 ]T ∈ N = {ω ∈ RN : L(ω) = L0 + Lk ωk−1 > 0}

(5-2)

k=1

with Li are given real symmetric matrices. To (in)validation the model MP , we need to answer the question: Do there exists a vector ω as in Eq. (5-2), and uncertainty k∆k∞ ≤ δ < 1 such that, the data obtained from the model is consistent with the real time data? If the answer is affirmative, then the model is not invalidated. Note that we can say that the model is certainly describes the system, rather the model correctly model the system for the given data. This is because that there might be some other data that can result negative outcome. In the section below, an Linear Matrix Inequality (LMI) method of answering the above question will be addressed.

Figure 5-2: LPV model (in)validation setup

5-2

LMI approach to LPV model (in)validation

In this section validation technique for LPV systems based on Fig. 5-2 is discussed. The method is mainly taken from the work of Mazzaro and Sznaier [13].

5-2-1

Preliminaries

Let denote by L∞ the Lebesgue space of complex valued matrix functions essentially bounded on the unit circle, such that: . kG(z)k∞ = sup σ ¯ (G(z)) (5-3) |z|=1

where σ ¯ denotes the largest singular value. Let H∞ be subspace of the functions in L∞ with abounded analytic continuation inside the unit disk. i.e. . kG(z)k∞ = sup σ ¯ (G(z)) (5-4) |z| 0 and L(ω) > 0

(5-15)

where . X(w∆ ) M (w∆ ) = Tw∆

TwT∆ Y (w∆ )

. X(w∆ ) = (TR Tu )T TR Tu + (TR Tu )T TS Tw∆ + TwT∆ TST TR Tu (5-16) . Y (w∆ ) =

1 δ2 I

− TST TS

−1

ωk = yk − PN [Go (Pk )] ∗ uk + PN [Q(Pk )] ∗ w∆,k with k = 0, 1, . . . , N − 1 and L(ω) is defined as in Eq. (5-2). Literature survey

S.K. Zegeye

38

5-3

Model Validation

Conclusion and results

In this chapter it has been discussed that (in)validation of system model is always done with a reference to real time measurement. But it can be done either to see if the model fits measurement data or to specify the uncertainty level incorporated in the model with respect the data. It has been indicated that the model validation technique do assert only that a model validated for certain data is correct model of the system for data set used. That is to say that we are incapable of invalidating the model with the available data. It has been also indicated that the LPV model of a system can be validated using LMI techniques, which can be solved using the available LMI toolbox. Furthermore, it becomes clear from the last section of the chapter that the model validation techniques takes into account the uncertainties defined in (Upper) Linear Fractional Transformation (LFT) form. Which gives insight in the result, the uncertainties that are incorporated in the model. Thus it can be concluded that model (in)validation techniques based on LMI frame work can suit to LPV systems.

S.K. Zegeye

Literature survey

Chapter 6 Concluding Remarks

For the reasons stated in Chapter 2, it has been clearly shown that Linear Time Invariant (LTI) modeling techniques can not capture the relevant dynamics of the wind turbines. The chapter clarified the challenges in developing simplified approximate model of the system. It has been stressed that the system is rather better approximated as an Linear Parameter Varying (LPV) system. It was reasonable, then, to discuss and elaborate LPV system identification. Chapter 3 gave a brief explanation about LPV system. The chapter also showed some of the techniques used in transforming nonlinear system to LPV form and its potential to capture the nonlinearity involved in systems. However, it has been concluded that there is no global LPV transforming techniques, rather the existing techniques are applied under certain condition. It has been described one of the works on LPV modeling rotational dynamics wind turbines. However, the modeling of flapping motion, nodding of the tower of the system remained intact. These dynamics which are coupled with the rotational motion of the system, have greater impact on the whole system. This imposes a challenge in obtaining an LPV model of the system which can adequately capture the dynamics. It is useful, to use real time data in obtaining an LPV model of the system, so that a comparison can be made with the model developed from the nonlinear equations of the system. Chapter 4 discussed the identification techniques so far developed. It elaborates the merits and demerits of each methods. As the scheduling parameters vector of wind turbine include both periodic and non-periodic elements, it inflicts a challenge to be tackled in the use of identification techniques so far developed. In Chapter 5, it was explained that the models developed has to be (in)validated. It has been shown that an Linear Matrix Inequality (LMI) approach can be used to determine the bound of the uncertainty involved in a system model. This is not a simple task, comparison of the models developed highly depend on the techniques to be used.

Literature survey

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40

Concluding Remarks

Therefore, there are mainly two open problems in LPV modeling of wind turbines. These could be stated as: 1) Develop an LPV model of wind turbines either using the transformation techniques so far developed or developing a new technique. 2) Obtain an LPV model of wind turbine by suiting the Identification techniques so far developed or develop a new technique. therefore, it is most likely that the thesis-project will not be a combination of tools and literature, thus handing in challenging project for the author. Based on the discussion between the supervisors of this project and the author of this report, a project assignment is traced out. This agreement can be found in Appendix A. This project agreement infers this literature survey, and leads the thesis-project to next step.

S.K. Zegeye

Literature survey

Bibliogarphy

[1] B. Bamieh and L. Giarre. Identification of Linear Parameter Varying models. Int. J. Robust Nonlinear Control, 12:841–853, 2002. [2] B. Bamieh and L. Giarre. LPV MODELS: Identification for Gain scheduling control. European Control Conference, 2001. [3] F.D. Bianchi, R.J. Mantz, and C.F. Christansen. Control of variable-speed Wind Turbines by LPV Gain Scheduling. WIND ENERGY, 7:1–8, 2004. [4] F.D. Bianchi, R.J. Mantz, and C.F. Christansen. Gain scheduling control of variablespeed wind energy conversion systems using quasi-LPV models. Control Engineering practice, 13:247–255, 2005. [5] L.H Carter and J.S. Shamma. Gain-Scheduled Bank-to-Turn Autopilot Design Using Linear parameter Varying Transformations. J. Guidance, control, and dynamics, 19(5): 1056–1063, September-October 1996. [6] K.D. Cock and B.D Moor. Subspace Identification Methods. KU Leuven, Department of Electrical Engineering Belgium. [7] F. Felici, J.W. Van Wingerden, and M. Verhaegen. Subspace identification of MIMO LPV systems using a periodic weight sequence. Delft Center for System and Control, Delft University of Technology, The Netherlands, 2006. [8] L. Giarre, D. Bauso, and B. Bamieh. LPV model identification for gain scheduling control: An application to rotating stall and surge control problem. Control Engineering Practice, 14:351–361, 2006. [9] L. Lee and K. Poolla. Identification of Linear Parameter-Varying Systems Using Nonlinear Programming. ASME Journal of Dynamic Systems, Measurement, and Control, 121(1):71–78, March 1999. Literature survey

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Bibliography

[10] D.J. Leith and W.E. Leithhead. Comments On the Prevalence of Linear Parameter Varying Systems, 1999. Tech. rep. Department of Electronic & Electrical Engineering, University of Strathclyde, Glasgow, Scotland. [11] F. Lescher and P. Zhao, J.Y.and Borne. Robust Gain Scheduling Controller for Pitch Regulated Variable Speed Wind Turbine. Studies in Informatics and Control, 14(4): 299–315, December 2005. [12] A. Marcos and J.G. Balas. Development of Linear-Parameter-Varying Models for Aircraft. J. Guidance, Control, and Dynamics, 27(2):218–228, March-April 2004. [13] C. Mazzaro and M. Sznaier. An LMI Approach to Model (In)Validation of LPV. Proceedings of American Control Conference, pages 5052–5057, June 25-27 2001. [14] D.P. Molenaar. Cost-effective design and operation of variable speed wind turbines. PhD thesis, Technical University of Delft, Delft, The Netherlands, 2003. [15] P.V. Overschee and B.D. Moor. N4SID: Subspace algorithms for the identification of combine detrministic-stochastic systems. Atomatica, 30(1):75–93, 1994. [16] G. Picci and T. Katayama. Stochastic realization with exogenous inputs and ’subspacemethods’ identification. Signal processing, 52:145–160, 1996. [17] F. Previdi and M. Lovera. Identification of a class of non-linear parametrically varying models. Int. J. Adapt. Control Signal Process, 17:33–50, 2003. [18] W.J. Rugh and J.S. Shamma. Research on gain scheduling. Automatica, 36:1401–1425, 2000. Survey paper. [19] J.Y. Shin. Quasi-Linear Parameter Varying Representation of General Aircraft Dynamics Over Non-Trim Region. Technical Report 2005-08, National institute of Aerospace, Hampton, Virginia, December 2005. [20] W. Tan. Application of Linear Parameter-Varying Control Theory. Master’s thesis, University of California, Berkeley, May 1997. [21] W. Tan, A.K Packard, and G.J. Balas. Quasi-LPV Modeling and LPV control of Generic Missile. Proceedings of the American Control Conference, pages 3692–3696, 2000. [22] Paul M.J. Van den Hof. System Identification. Delft University of Technology, January 2006. Lecture notes of SC4110. [23] V. Verdult. Nonlinear System Identification: A State-Space Approach. PhD thesis, University of Twente, Enschede, The Netherlands, 2002. [24] V. Verdult and M. Verhaegen. Subspace identification of multivariable linear parametervarying systems. Automatica, 38:805–814, 2002. [25] V. Verdult and M. Verhaegen. Kernel methods for subspace identification of multivariable LPV and bilinear systems. Automatica, 41:1557–1565, 2005. S.K. Zegeye

Literature survey

Bibliography

43

[26] V. Verdult, N. Bergoer, and M. Verhaegen. Identification of fully parametrized linear and non-linear state-space systems by projected gradient search. In In Preprint of the IFAC Symposium on System Identification (SYSID), pages 737–742, Rotterdam, The Netherlands, August 2003. [27] V. Verdult, M. Lovera, and M. Verhaegen. Identification of linear parameter-varying state space models with application to helicopter rotor dynamics. Int. J. Control, 77 (13):1149–1159, 2004. [28] M. Verhaegen, V. Verdult, and N. Bergboer. Filtering and System Identification: An Introduction to using Matlab Software. Delft University of Technology, June 21 2003. Lecture notes for the course SC4040. [29] WWEA. World Wind Energy Association. http://www.wwindea.org.

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S.K. Zegeye

Bibliography

Literature survey

Appendix A Formal MSc-Project Agreement

Title and Project Description Title : Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch-Regulated Wind Turbine. Project Description: The project concentrates on the development of quasi-Linear Parameter Varying (LPV) model of Wind Turbines using LPV transformation and identification techniques. Hereafter is given the project description of the MSc-thesis. It is divided into three phases of one or more steps.

Phase 1: LPV transformation of the nonlinear equations 1. Selection of scheduling variables: The literature survey provides nonlinear model of wind turbines. Selection of scheduling variables is a crucial step in transforming the nonlinear model to LPV form. This step will result in: • A chapter on modeling of Wind Turbine and selection of scheduling variables. • List of physical parameters of Wind Turbine (NM80)

• MATLABr /Simulinkr model of the nonlinear system for simulation 2. Selection of LPV transformation techniques: The literature provides couple of techniques that can be used for the transformation of nonlinear systems to LPV form. However, the methods are problem dependent. Thus in this step techniques that can better approximate the system in LPV form will be selected. This will result in: • a Chapter on LPV transformation of Wind Turbines.

• MATLABr /Simulinkr model of the LPV model and its simulation results. Literature survey

S.K. Zegeye

46

Formal MSc-Project Agreement

Phase 2: LPV identification 1. Selection of Identification techniques: In the literature it has been shown that there exist different methods for the identification of LPV systems. But, all the methods have some disadvantages. Here, in this step a method(or combination of methods) which, better suits for the identification of wind turbines will be selected. Hence this will result in: • a chapter on identification of wind turbines.

• MATLABr /Simulinkr model and simulation results.

Phase 3: Model validation A model validation technique based on Linear Matrix Inequality (LMI) approach is going to be applied. This phase will result in: • a chapter on model validation of LPV wind turbine model. • Determining the uncertainty involved in the model.

Project Team MSc-Student: MSc-Thesis advisor: MSc-Thesis daily supervisor: Chairman examination committee: Others:

S.K. Zegeye Prof. dr. ir. M.Verhaegen Ir. J.W. Van Wingerden Prof. dr. ir. M.Verhaegen

Planning of course of project Planned Planned Planned Planned

date of starting: date of literature colloquium: month of introductory colloquium: month of examination:

July 15, 2006 November 24, 2006 January 2007 July 2007

Additional Requirements The entire project-team will meet approximately once a month. ir. J.W. van Wingerden and the Msc-student will meet once every two weeks. Furthermore, the Msc-student will provide the other members of the project team with a project report at least 1 day prior the next meeting. S.K. Zegeye

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47

Other issues (if applicable) • List of required facilities (like laboratory equipments): will be discussed when required. • Budgeting (if extra expenses are expected): will be discussed when required.

Signature and Date

........................./...../...../.......... MSc-student

........................./...../...../.......... MSc-Thesis Advisor

........................./...../...../.......... DCSC-Management

Literature survey

S.K. Zegeye

Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch ...

Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch ...

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Time-Varying Parameter Vector Autoregressions - SSRN

Masters Thesis: Quasi-Linear Parameter Varying ... - CiteSeerX

Time-varying parameter error correction models: the

Robust Multivariable Linear Parameter Varying ... - Google Sites

VARYING COEFFICIENT REGRESSION MODELING ...

Modeling pitch errors of Japanese intonational

Quantitative Modeling of Pitch Accent Alignment

Modeling and Prediction of Time- Varying ...

parameter optimization of pitch controller for robust frequency ... - Core

aperture-varying autoregressive modeling of ...

Linear Parameter Varying Control of Permanent Magnet Synchronous ...

Review of Parameter Estimation Techniques for Time-Varying ...

Solution of Parameter-Varying Linear Matrix Inequalities in Toeplitz Form

Constrained Time-Optimal Control of Linear Parameter-Varying Systems

A Time-Varying Parameter Model of Inflation in Indiaâ

Robust and Linear Parameter-Varying Control of ... - tub.dok

The Effects of Varying Parameter Values and ... - Semantic Scholar

Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch ...