Skip to main content

ORIGINAL RESEARCH article

Front. Control Eng., 06 February 2023
Sec. Adaptive, Robust and Fault Tolerant Control
This article is part of the Research Topic Reliable Modeling, Simulation, Identification, Control and State Estimation for Dynamic Systems with Uncertainty View all 7 articles

Global Versus Local Lyapunov Approach Used in Disturbance Observer-Based Wind Turbine Control

Updated
Eckhard Gauterin
Eckhard Gauterin*Florian PschkeFlorian PöschkeHorst SchulteHorst Schulte
  • Control Engineering Group, Department of Engineering I, University of Applied Sciences HTW Berlin, Berlin, Germany

This contribution presents a Lyapunov-based controller and observer design method to achieve an effective design process for more dedicated closed-loop dynamics, i.e., a maximal flexibility in an observer-based controller design with a large consistency in desired and achieved closed-loop system dynamics is intended. The proposed, pragmatic approach enhances the scope for controller and observer design by using local instead of global Lyapunov functions, beneficial for systems with widely spaced pole locations. Within this contribution, the proposed design approach is applied to the complex control design task of wind turbine control. As the mechanical loads that affect the wind turbine components are very sensitive to the closed-loop system dynamic, a maximum flexibility in the control design is necessary for an appropriate wind turbine controller performance. Therefore, the implication of the local Lyapunov approach for an effective control design in the Takagi-Sugeno framework is discussed based on the sensitivity of the closed-loop pole locations and resulting mechanical loads to a variation of the design parameters.

1 Introduction

The mechanical loads, affecting a wind turbine (WT), are very sensitive to the closed-loop system dynamics. Hence, for the design of an appropriate WT controller, a maximal flexibility is necessary to mitigate the resulting, mechanical loads. For this purpose, model-based and automated controller optimisation procedures are recommended. Until now, the authors achieved the intended flexibility with decomposed, structural dynamic models of the wind turbine (e.g., (Pöschke et al., 2020)) and with an observer-based controller structure (Gauterin et al. (2014), Pöschke et al. (2019)), whose separately designed observer and controller are based on a common and global Lyapunov approach, respectively. With the local Lyapunov approach, conceived in the outlook of (Pöschke et al., 2022) and presented in this work the first time, the evolution of the design process with an increased controller flexibility and improved consistency is proceeded by the implementation of a more effective controller design procedure.

In control theory, the Lyapunov approach (Lyapunov, 1992) is utilised for controller synthesis, i.e., to analyse the stability of closed-loop systems and simultaneously providing the related controller gains. Within a model-based control design, the Lyapunov approach enables an automated design process.

As most real world systems are characterised by complex, nonlinear dynamics, techniques to analyse stability and dynamical characteristics are needed. For this, the Takagi-Sugeno (TS) framework (Takagi and Sugeno, 1985) may be used, that describes a nonlinear system as convexly blended linear submodels (Ai, Bi) (Tanaka and Sano, 1994a). A thorough discussion on the TS methodology is given in (Tanaka and Wang, 2001) and with a focus on observer-based methods in (Lendek et al., 2010). To form the TS model structure, the individual linear submodels (Ai, Bi) may be gained with the sector nonlinearity approach (Tanaka and Sano, 1994b), which yields an exact representation of the nonlinear system, or by linearisation (Johansen et al., 2000). As the identification and analysis of numerically derived linearised models is an established approach to investigate control properties in the wind turbine application (Bossanyi, 2000), the linearisation approach is used within this proceeding, too. To facilitate this, aero-elastic simulation programs like NREL FAST (Jonkman and Buhl, 2005; Jonkman, 2016) provide a linearisation feature to easily obtain the used matrices Ai, Bi as discussed in (Jonkman and Jonkman, 2016). The resulting TS model allows for the analysis of the dynamical properties and stability of both, the open- and closed-loop system.

Using the inequality of the Lyapunov approach on these convexly blended combinations of linear submodels in its matrix formulation, linear matrix inequalities (LMI, (Boyd et al., 1994)) are derived that describe the stability condition of the system dynamics (Tanaka and Wang (1997), Tanaka and Wang (2001), Lendek et al. (2010)). Additionally, performance constraints, in form of physically interpretable pole regions, can be specified and formulated in terms of LMIs (Chilali and Pascal, 1996). The combination of the stability condition with the performance constraints forms a collection of LMIs, which can be efficiently solved with numerical LMI solvers (VanAntwerp and Braatz, 2000). The LMIs’ solution space and thereby the solution’s conservativeness is restricted by the stability condition and the number of performance constraints taken into account. However, the conservativeness may be influenced, e.g., by the way the TS model is constructed and/or the use of relaxations in the resulting LMI (Tanaka et al. (1998), Tanaka and Wang (2001)).

For an observer-based controller, a separated controller and observer design can be used. The corresponding separation principle is also valid for observer-based TS controllers (Yoneyama et al. (1998), Ma et al. (1998)), but does not necessarily hold for parameter uncertainties or stochastic noise. Therefore, design procedures are investigated, which account for these restrictions (Zemouche et al., 2016), (Rauh et al., 2021). As an overall observer and controller design often results in conservative controller and control objective performances, respectively, pragmatic design syntheses are intended for real world application, rather than a guaranteed overall stability of the observer-based controller. From engineering point of view, a controller with guaranteed, overall system stability does not ensure stability for the closed-loop real world system dynamics, as the controller design model cannot cover all uncertainties (e.g., resulting from unpredicted or non-modelled environmental influences).

With the proposed local Lyapunov approach, a pragmatic controller design procedure is introduced, that utilises a separated controller and observer design and reduces the stability conditions from global stability conditions (of the nonlinear system within the defined operational range of the TS description) to local stability conditions (at each considered operating point, i.e., small-signal stability). With the resulting reduction of the number of parallel to be solved LMIs, the constraints for the solver are reduced and thereby the flexibility in assigning desired pole locations of the closed-loop system is increased, i.e., the pole region bound modifiability and flexibility, respectively is significantly improved. The less conservative task for the LMI solver enables the specification of tighter performance constraints (e.g., smaller pole regions) and reaches an increased consistency in desired and achieved closed-loop dynamics. The increased consistency and increased flexibility, are hereinafter denoted as the general objective of the local Lyapunov approach (see Section 2.3). Hence, the pragmatic local Lyapunov approach provides an enhanced design scope for real world applications and results in a control design method, which is highly effective in posing more dedicated closed-loop dynamics, especially beneficial for systems with widely spaced pole locations.

Within this contribution the concept of a local Lyapunov approach is described in detail the first time and applied for wind turbine (WT) systems, characterised by widely spaced pole locations of the open-loop system, due to the divergent stiffness, damping and inertia of the main components, like rotor blades and tower. Also for these challenging system dynamics, WT controllers in a TS framework have demonstrated to be capable for energy yield optimisation, load mitigation and active power reduction (Pöschke et al., 2020) or fault tolerant control (Georg (2015), Schulte and Gauterin (2015)). The utilised TS WT system model is achieved by linearising an elaborated WT simulation model (the NREL FAST 5 MW reference WT model (Jonkman et al., 2009)). Thereby, all mechanical couplings between the WT main components’ degree of freedom are neglected to increase the flexibility for the controller design. That is, the overall, decomposed TS model comprises decoupled TS WT main component models (with each TS WT main component model consisting of convexly blended linear submodels) that are achieved from linearisation and afterwards combined to the overall, decomposed TS WT model.—As a precise wind speed measurement is hardly feasible with conventional, cost-effective anemometers on WT (e.g., due to high turbulences induced by the rotor (Ostergaard et al., 2007) and the significant variation of the local wind speeds within the enormous size of the rotor swept area), advanced, expensive sensors (like LiDaR systems (Schlipf et al., 2010)) and observer techniques are investigated for the WT application ((Ma et al. (1995), Ostergaard et al. (2007), Jena and Rajendran (2015), Gauterin et al. (2015)). Within this contribution, a TS disturbance observer is used for an estimation of the unknown wind speed as premise variable, enabling a premise variable scheduled feedforward actuation for disturbance attenuation (Gauterin et al., 2014), while the—also premise variable scheduled—feedback-controller just compensates the control signal deviations resulting from model-uncertainties. To evaluate the observer-based TS feedforward-feedback controller achieved with the local Lyapunov observer design approach, the NREL FAST 5 MW reference WT model is used for WT operation simulation.

The paper is organised as follows: In Section 2 Method, the TS framework (Section 2.1), the global and local Lyapunov approach (Section 2.2 and Section 2.3) and its application to WT control (Section 2.4) with additional performance constraints (Section 2.5) is introduced. Section 3 Simulations and Results describes the simulation design (Section 3.1) and the achieved results (Section 3.2). Finally, in Section 4 Discussion the achieved results are assessed and a Conclusion is drawn in Section 5.

2 Method: Global and Local Lyapunov Approach

In this Section the Takagi-Sugeno (TS) framework (Section 2.1) and the global Lyapunov approach are presented (Section 2.2). In Section 2.3 the local Lyapunov approach is introduced to system models described in the TS framework. Its application for wind turbine control is presented in Section 2.4 and additional controller design constraints to define pole regions with regard to wind turbine application are described in Section 2.5.

2.1 System Model, Controller and Observer in Takagi-Sugeno Framework

2.1.1 System Model

In this proceeding, a nonlinear system ẋ̲=f(x̲,u̲) is described in the Takagi-Sugeno framework by convexly blended linear submodels, based on Taylor linearisation of the nonlinear system.

For the wind turbine (WT) application, the plant model is generated by linearising the nonlinear system model at Nr steady state operating points (piecewise equidistant regarding the disturbing wind speed v), resulting in a set of Nr linear submodels (Ai, Bi) (with i ∈ [1, imax] and imaxNr). These submodels comprise the state matrices Ai and input matrices Bi for the state vector x̲ and input vector u̲ and their steady state values x̲c,i and u̲c,i. In wind turbine application the nonlinearity mainly results from the aerodynamics of the rotor blades.

The linear submodels (Ai, Bi) are blended in a convex sum

ẋ̲=iNrhiz̲Aix̲x̲c,i+Biu̲u̲c,iwithy̲=Cx̲andiNrhiz̲=1with0hiz̲1(1)

with the help of membership functions hi(z̲), which depend on the premise variable z̲. Within this proceeding, the membership functions hi(z̲) are defined by triangular-shaped weighting functions wk,l (zk) (with the actual premise variable zk(t) and zk, respectively between the discrete linearisation points OPi for zkl1,zkl and zkl+1, see Figure 1) for each of the k ∈ [1, kmax] premise variables zk (discretised in l (zk) ∈ [1, lmax (zk)] linearisation points with

wk,lzk=zkzklzklzkl1ifzkl1<zkzkl1zkzklzkl+1zklifzkl<zkzkl+10else.(2)

FIGURE 1
www.frontiersin.org

FIGURE 1. Illustrative example for pyramid-shaped membership functions hi(z̲)=hi(z1,z2) with the two premise variables z̲=[z1,z2]T (formed by the triangular-shaped weighting functions w1,l (z1) and w2,l (z2), which are discretised in l (zk)∈[1, lmax (zk)] linearisation points; see Eq. 2 and exemplary visualisation of the eight direct adjacent steady Operating Points (and linearisation points, respectively) OPi1lmax(z1) to OPi+1+lmax(z1) adjoining OPi with their membership functions (see the overlapping parts of the nine coloured pyramids) hi1lmax(z1)(z1l1,z2l1) to hi+1+lmax(z1)(z1l+1,z2l+1) (with hi(z̲)0 within z1l2z1z1l+2 and z2l2z2z2l+2 for each of the nine steady operating points, while hi(z̲)=0 holds for all the other steady operation points and z̲, respectively; with i =(l (z2)−1)lmax (z1)+ l (z1))

The weighting functions wk,l (zk) are combined to form the (imax)Nr=k=1kmaxlmax(zk) membership functions hi(z̲) (with i ∈ [1, Nr]):

i=1Nrhiz̲t=k=1kmaxl=1lmaxzkwk,lzk.(3)

In this contribution, the membership functions hi(z̲) just blend the i submodels, which are direct adjacent to the actual operation point (with hi(z̲)0), while the membership functions of all other models are set to zero (i.e., hi(z̲)=0), as described in e.g., (Pöschke et al., 2020) and illustrated in Figure 1.

In WT application, it is advantageous to define the reconstructed wind speed v̂ as the premise variable z̲v̂.

Within this contribution individual input-matrices Bi, e.g., depending on the wind speeds v and rotor rotation speed ωR, and a common output-matrix CiC, just describing the measurable states y̲, are supposed.

2.1.2 Parallel-Distributed-Compensation-Controller With Feedforward Actuation

In the TS framework, a state-space controller u̲=Kx̲ with convexly blended state-feedback gains Kj—so-called Parallel Distributed Compensation (PDC) Controller (Wang et al. (1995), Tanaka and Wang (1997))—is usually used:

u̲=jNrhjz̲Kjx̲x̲c,j.(4)

In Eq. 4 and the following, it is assumed, that ẑ̲z̲ holds (as explained in Section 2.4.4).

If the disturbance attenuation is realised with a feedforward actuation, the PDC controller (Eq. 4) is extended to

u̲=jNrhjz̲Kjx̲x̲c,ju̲FB+jNrhjz̲u̲c,ju̲FF=jNrhjz̲Kjx̲x̲c,ju̲c,j.(5)

With the feedforward signal u̲FF the disturbance is attenuated, while the feedback signal u̲FB compensates control errors resulting from design model uncertainties.With Eq. 5 and iNrhi(z)=1 in Eq. 1 the closed-loop dynamics (for i individual input matrices Bi, see Section 2.1.1)1 is described by

ẋ̲=iNrhiz̲Aix̲x̲c,i+BijNrhjz̲Kjx̲x̲c,ju̲c,ju̲c,i=iNrjNrhiz̲hjz̲AiBiKjx̲x̲c,i.(6)

2.1.3 Takagi-Sugeno Observer

For nonlinear systems ẋ̲=f(x̲,u̲) a weighted combination of linear Luenberger observers (Luenberge, (1971), Luenberger (1964)) can be used, which is denoted as Takagi-Sugeno Observer (TSO). The TS observer is obtained from the TS system Eq. 1 by introducing the output error-feedback term Li(y̲ŷ̲) (Tanaka and Sano, 1994a). For a common output matrix CiC (see Section 2.1.1) it holds2:

x̲̂̇=iNrhiz̲Aix̲̂x̲c,i+Biu̲u̲c,i+iNrhiz̲Liy̲ŷ̲ŷ̲=Cx̲̂=iNrhiz̲Aix̲̂x̲c,i+Biu̲u̲c,i+LiCx̲x̲̂(7)

with the reconstructed states x̲̂, the reconstructed outputs ŷ̲ and the error-feedback gains Li.

For the error dynamics ė̲

e̲x̲x̲̂ė̲=ẋ̲x̲̂̇(8)

it follows with Eqs 7, 1 in Eq. 8 (with the common output matrix C, see Section 2.1.1 and Section 2.2.2)2:

ė̲=iNrhiz̲Aix̲x̲̂+LiCx̲x̲̂=iNrhiz̲Ai+LiCx̲x̲̂e̲.(9)

Within this contribution (and the previously published works in (Gauterin et al., 2014) and (Pöschke et al., 2020)) the TS-observer is implemented to reconstruct the disturbing wind speed v̂, that is used as the premise variable z̲, scheduling the feedforward and feedback signal (see Eq. 5 and Figure 3 with z̲=v̂). Therefore, the TS-observer does not represent a typical disturbance observer for explicit disturbance rejection, rather than a premise observer to reconstruct the disturbance signal v̂ as premise variable z̲, that is used to influence the controllable system inputs with the premise variable governed control signal scheduling (see explanation for Eq. 5 and Section 2.4.3). For the wind turbine application, the premise variable z̲ often comprises the reconstructed, disturbing wind speed v̂ (within this contribution, the premise variable consists just of the reconstructed wind speed v̂, i.e. z̲=v̂ holds), therefore the premise observer is also denoted as disturbance observer in the following (see also the observer classification in Section 2.4.3).

2.2 Global Lyapunov Approach Based Takagi-Sugeno Controller and Observer Design

The state-feedback gains Kj and error-feedback gains Li are achieved from the stability condition, based on a Lyapunov approach: Assigning a simple, quadratic Lyapunov function, the following global stability condition yields

VX̲TPX̲>0V̇=Ẋ̲TPX̲+X̲TPẊ̲<0withPT=P0(10)

for the common and global, respectively, symmetric, positive definite matrix P ≻ 0 and the system-states (X̲,Ẋ̲)(x̲,ẋ̲) or error-states (X̲,Ẋ̲)(e̲,ė̲). That is, if the common, symmetric and positive definite matrix P exists, which holds the stability condition for the common and global, respectively, quadratic Lyapunov function V Eq. 10, the system is globally asymptotically stable and the state- and error-feedback-gain Kj and Li can be derived from P as shown in the following two Subsection 2.2.1 and Subsection 2.2.2, further information given e.g., in (Lendek et al., 2010), (Wang et al., 1996) and (Tanaka and Sugeno, 1992).

Hereinafter, the Lyapunov approach Eq. 10 is denoted as the global Lyapunov approach.

2.2.1 Global Controller

LMI derivation

With Eq. 6 in Eq. 10 (x̲X̲) it follows (for the individual input-matrices Bi)1:

V̇=x̲TiNrjNrhihjAiTPKjTBiTP+PAiPBiKjx̲<0.(11)

With the pre- and postmultiplication P−1 ⋅ □ and □⋅ P−1, the thereby necessary substitution XP1(=Eq.10X) and the introduction of the slag parameter Mj = Kj X (to avoid the bilinear term Kj X within the resulting inequality) the following inequality for the Lyapunov function dynamics is derived from Eq. 11:

V̇=x̲TiNrjNrhihjXAiTMjTBiT+AiXBiMjx̲<0.(12)

The inequality Eq. 12 is solvable with a LMI-solver, if a discrete number of LMIs is derived from Eq. 12. Therefore, the convex properties of the membership functions hi hj are exploited to derive the following LMI set with the intended discrete number of LMIs:

XAiTMjTBiT+AiXBiMj0.(13)

If the LMI set Eq. 13 is solvable, i.e., a positive definite matrix X=P1 with P ≻ 0, see Eq. 10) exists, the Lyapunov approach Eq. 10 (and its derivative Eq. 12) is fulfilled and the closed-loop system’s stability is guaranteed.

Controller design procedure

Once a common matrix P and the slag parameter Mj are found with the LMI solver, the state-feedback gains Kj are defined by Kj = Mj P. That is, for individual input matrices Bi the state-feedback gain Kj of the jth submodel and subcontroller, respectively, is designed in a way, that the subcontroller holds the LMI Eq. 10 and LMI set Eq. 13, the latter combining1 the closed-loop dynamic of the jth submodel with the dynamic of all other or the direct adjacent submodels.

Number of LMIs

For the discrete number of combined1 LMIs in the LMI set Eq. 13 it holds: nglobal,allLMI,Cntrl=jmaximax=Nr2 (with imax = jmax = Nr). This number can be reduced to nglobal,adj.LMI,Cntrl<nglobal,allLMI,Cntrl, if just submodels, that are direct adjacent to the jth submodel (see Section 2.1.1), are blended. Additionally, the stability condition Eq. 10 comprises a single LMI. That is, for the controller design with individual input matrices Bi the total LMI set consists of nglobal,all∕adj.LMI,Cntrl+1 LMIs. For the global Lyapunov approach with the global and single P-matrix, respectively this total LMI set is solved with a single execution of the LMI solver, i.e., the nglobal,all / adj.LMI,Cntrl+1 LMIs are solved simultaneously.

2.2.2 Global Observer

LMI derivation

With Eq. 9 in Eq. 13 (e̲X̲) it follows (for the common output-matrix CiC)2:

V̇=e̲TiNrhiAiTPCTLiTP+PAiPLiCe̲<0.(14)

With the introduction of the slag parameter Ni = P Li (to avoid the bilinear term P Li within the resulting inequality) the following inequality for the error dynamics is derived from Eq. 14:

V̇=e̲TiNrhiAiTPCTNiT+PAiNiCe̲<0.(15)

Eq. 15 holds (as explained for Eq. 12), if the LMI set

AiTPCTNiT+PAiNiC0(16)

is satisfied, i.e., a positive definite matrix P ≻ 0 (see Eq. 10) exists, so that the Lyapunov approach Eq. 10 (and its derivative Eq. 15) is fulfilled and the error system’s stability is guaranteed.

Observer design procedure

Once a common matrix P and the slag parameter Ni are found with the LMI solver, the error-feedback gains Li are defined by Li = P−1 Ni. That is, for a common output matrix CiC the error-feedback gain Li of the ith submodel and subobserver, respectively is designed in a way, that the subobserver holds the LMI Eq. 10 and the LMI set Eq. 16, the latter just comprising the error dynamics of each single submodel (Ai, Bi) in an individual and single LMI, respectively (and not combining the dynamics of all or direct adjacent submodels).

Number of LMIs

For the discrete number of individual LMIs in the LMI set Eq. 16 it holds: nglobalLMI,Obs=imax=Nr. As Eq. 16 defines a single LMI for each submodel and subobserver, respectively, there is no need to consider direct adjacent submodels in the observer design with a common output matrix C. Additionally, the stability condition Eq. 10 comprises a single LMI. That is, for the observer design with a common output matrix C the total LMI set consists of nglobalLMI,Obs+1 LMIs. For the global Lyapunov approach with the global and single P-matrix, respectively this total LMI set is solved with a single execution of the LMI solver. i.e., the nglobalLMI,Obs+1 LMIs are solved simultaneously.

2.3 Local Lyapunov Approach for Takagi-Sugeno Controller and Observer Design

For the proposed local Lyapunov approach the local stability condition

ViX̲TPiX̲>0Vi̇=Ẋ̲TPiX̲+X̲TPiẊ̲<0withPiT=Pi0andi1,Nr(17)

holds for the individual and local, respectively, symmetric, positive definite matrix Pi ≻ 0 and the system-states (X̲,Ẋ̲)(x̲,ẋ̲) or error-states (X̲,Ẋ̲)(e̲,ė̲). Compared to the common and global Lyapunov-approach Eq. 10, a number of Nr individual Lyapunov functions Vi (with i ∈ [1, Nr]) are used instead of one common Lyapunov function V (in the global Lyapunov approach Eq. 10). That is, for each submodel (Ai, Bi) an individual Lyapunov function Vi is defined.

Accordingly, the LMI derived for the closed-loop dynamics Eq. 13 and error dynamics Eq. 16 is simplified, as the convex blending becomes obsolete, if the Lyapunov stability condition is defined for each submodel individually (compare Eq. 18 with Eq. 12 and Eq. 20 with Eq. 15).

2.3.1 Local Controller LMI

For the local Lyapunov approach the inequality derived for the closed-loop dynamics (with (X̲,Ẋ̲)(x̲,ẋ̲)) results in

V̇=x̲TXiAiTMiTBiT+AiXiBiMix̲<0withXiPi1andMi=KiXi.(18)

The inequality (Eq. 18) holds, if the LMI

XiAiTMiTBiT+AiXiBiMi0(19)

is satisfied, i.e., individual and local, respectively positive definite matrices Xi exist, which fulfill Eq. 19.

2.3.2 Local Observer LMI

The same simplification holds for the error dynamics, i.e. for the local Lyapunov approach the inequality derived for the error dynamics (with (X̲,Ẋ̲)(e̲,ė̲)) results in:

V̇=e̲TAiTPiCTNiT+PiAiNiCe̲<0withNi=PiLi.(20)

The inequality (Eq. 20) holds, if the LMI

AiTPiCTNiT+PiAiNiC0(21)

is satisfied, i.e. individual and local, respectively positive definite matrices Pi exist, which fulfill Eq. 21.

2.3.3 Local Controller and Observer Design Procedure

Once the individual matrix Pi and the slag parameter Mi (for local controller design) or Ni (for local observer design) is found with the LMI solver, the state-feedback gain Ki or error-feedback gain Li is defined by Ki = Mi Pi or Li=Pi1Ni. That is, the state-feedback gain Ki or error-feedback gain Li of the ith submodel and subcontroller or subobserver, respectively is designed in a way, that the subcontroller or subobserver holds the LMI Eq. 17 and the LMI set Eq. 19 or Eq. 21, the latter just comprising the state or error dynamics of each single submodel (Ai, Bi) in an individual and single LMI, respectively (and not combining the dynamics of all or direct adjacent submodels).

2.3.4 Number of LMIs

As the local Lyapunov approach simplifies and reduces, respectively the set of combined LMIs (for the local Lyapunov approach the LMI set comprises just two LMIs: LMI Eq. 19 or Eq. 21 and the LMI of the positive definite matrix Pi ≻ 0, see Eq. 17), the parallel and simultaneously, respectively to be solved LMIs are reduced (from nglobal,all∕adj.LMI,Cntrl+1>nglobalLMI,Obs(=Nr+1) to nLocalLMI,Cntrl∕Obs=2 LMIs), too. Thereby, the LMI solver is executed Nr times consecutively for the local Lyapunov approach (due to the individual Pi-matrix in Eq. 17), while the LMI solver for the global Lyapunov approach is executed just once, because of the common P matrix in Eq. 10. Therefore, the flexibility of the LMI solver in assigning desired pole locations of the closed-loop system is increased for the local Lyapunov approach. The less conservative task for the LMI solver enables the specification of tighter performance constraints for the local Lyapunov approach, e.g., smaller pole regions (by implementing pole region constraints in form of additional LMIs, see Section 2.5), and results in more flexible and consistent specifications of the desired closed-loop dynamics, denoted as the general objective of the local Lyapunov approach.

2.3.5 Subsequent, Global Lyapunov Stability Analysis

Note: With the local Lyapunov approach and the resulting LMIs Eq. 19 and Eq. 21, just the small signal stability is evaluated, i.e., considering the operating point depicted in the linear submodel, if a positive definite matrix Pi exists. To ensure the global stability of the convexly blended submodels, a final stability analysis has to be performed, enveloping all combined closed-loop submodels. This final stability analysis is executed in the subsequent, global Lyapunov stability analysis described in Section 2.4.4.

2.4 Local Lyapunov Approach in Wind Turbine Control Application

2.4.1 Structural Dynamical Design Model

For the wind turbine (WT) controller design, the authors use simplified structural dynamical models, based on lumped-masses and joined with discrete spring- and damper-elements (e.g., (Bianchi et al., 2007), (Georg, 2015) or (Pöschke et al., 2020)). These models comprise just the essential and costly wind turbine components rotor, drive train and tower. Within this contribution, the rotor blade dynamic is represented, while the tower dynamic is neglected (see Figure 2). The rotor model is composed of the rotor Blades, rotating as one rigid body in the rotor plane (denoted with rotor rotation (speed) ωR in Figure 2) and translating with xB in and against the wind direction (denoted by the subscripted index B). The drive train is represented by its rigid body Rotation (denoted by the subscripted index R).

FIGURE 2
www.frontiersin.org

FIGURE 2. Structural dynamical design model of a wind turbine (in side- and front-view) with the two degree of freedom: • blade translationxB (in up- and downwind direction) and • drive train dynamic, i.e., rotor and generator rotation (speed)ωR for the rigid body model of the rotor and drive train. The blue depicted ball bearings enable just the translation xB of the discrete rotor model in relation to the discrete drive train model (i.e., in wind direction), while the red depicted bearings enable just the drive train rotation ωR along its horizontal axis and suspend any other coupling of drive train rotation and tower translation. Linear, single headed arrows represent translations, forces or translatory spring- and damper-elements, while linear, double headed arrows represent rotations. The parameters mK, cK and dK represent the discrete component masses, discrete spring coefficients and discrete damping coefficients of the discrete component model K (with K=B(lade) or K=R(otation) of the drive train.)

After linearising an elaborated WT simulation model for i stationary operating points OPi (with i ∈ [1, Nr]), the controller design submodels (AiB/R,BiB/R) are composed with the linearisation coefficients. Thereby, mechanical couplings between those components are specified or neglected, as explained in Section 2.4.4. The specification of the design submodels (AiB/R,BiB/R) are given in Tables A2, A3 in the appendix.

2.4.2 Wind Turbine Control Objectives, Loading and Operation Concept

In wind turbine (WT) application the controller intends—besides energy yield optimisation—for mechanical load mitigation, due to vast environmental loads acting on the complete WT structure. Besides the ultimate, mechanical loads (that occur for single moments and the corresponding ultimate stress must not exceed the material strength), fatigue, mechanical loads have to be examined. Those fatigue loads, also denoted as Damage Equivalent Loads (DEL, i.e., a mean amplitude with the equivalent damaging effect like constantly or stochastically changing amplitudes, e.g., resulting from turbulent wind time series), characterise the damaging effect resulting from load cycles, occurring all over the components’ lifetime (Clem ens et al., 2020).

Ascribed to the generator characteristics, two operating modes have to be distinguished: In the partial load range, the generator is operated below rated power and the energy yield is optimised by generator torque TG control, i.e., the generator torque TG is one of the two actuating signals. In full load range, the generator is operated at rated power. Therefore, the energy extraction from the inflow with the WT rotor is restricted to rated generator power by the pitch angle β control, influencing the blade aerodynamics and mechanical torque TR generation of the rotor by rotating the complete rotor blade along its longitudinal axis (see Figure 2). Hence, the pitch angle β is the second actuating signal of a wind turbine controller, i.e., u̲=[β,TG]T holds. In this contribution, a collective pitch control algorithm is utilised, i.e., all blades are actuated with the same pitch angle. In Appendix Table A1, the control signals u̲c,i=[βc,i,TG,c,i]T for all 27 stationary operating points in partial and full load range are listed.

2.4.3 Implemented Wind Turbine Controller Structure

Within this contribution an observer-based feedforward-feedback controller in TS framework is utilised and applied to wind turbine control.

For the feedforward-feedback controller the extended parallel distributed compensation (PDC) controller Eq. 5 is used.

To attenuate the effect of the disturbing wind speed, a disturbance-observer is used (see Section 2.1.3 and Figure 3) to determine the rotor effective wind speed veff. For the disturbance observer, the WT main component model (see Section 2.4.1 and Section 2.4.4) with the highest disturbance sensitivity is used, i.e., for wind turbine systems the downwind and upwind blade deflection xB3 is most sensitive to disturbances caused by wind speed fluctuations, due to the very low bending stiffness of the blades (in normal direction to its profile chord). Therefore, the disturbance observer reconstructs the disturbing, effective wind speed v̂veff and blade translation speed x̂̇B from the blade translation error exB determined from the measured and reconstructed blade translation exB=xBx̂B. For this purpose, the disturbance observer design model is augmented by a simple wind model (Ekelund, 1994), expanding the state vector x̲̂B with the reconstructed wind speed v̂ (and disturbance signal d̲v̂, respectively) x̲̃̂B=[x̲̂B,v̂]T and augmenting the state space model Eq. 7 and Eq. 9 accordingly (with Ãi,B̃i and C̃; see also (Pöschke et al., 2020)). Therefore, the used TS disturbance observer, resulting from the integration of a disturbing model in the plant model and augmenting the state vector x̲̂B with the disturbing wind state v̂, is classified as an Extended State Observer (see (Li et al., 2014)), that is represented in the Takagi-Sugeno framework.

FIGURE 3
www.frontiersin.org

FIGURE 3. Block digram of the observer-based Takagi-Sugeno wind turbine controller.

The reconstructed wind speed v̂ is defined as premise variable z̲, scheduling the nonlinearity of the system and utilised for the membership function hi(z̲) of the feedforward signal u̲FF and feedback signal u̲FB in Eq. 5. Therefore, the disturbance observer does not intend for a disturbance rejection, rather than for an observer-based controller scheme with disturbance reconstruction, influencing the controllable system inputs (see also Section 2.1.3).

Within this contribution, triangular-shaped4 membership functions hi(z) are used, just blending directly adjacent submodels (Ai, Bi) (see Figure 1 and the explanations in Section 2.1.1). Because of the reduced number of (triangular) membership functions (i.e., just the direct adjacent membership functions are taken into account and not all membership functions), also the number of submodels included in the feedback controller and observer design—based on the global Lyapunov approach—is reduced (see Section 2.3) (Note: For the local Lyapunov approach an individual Lyapunov function Vi is defined for each submodel (Ai, Bi), because of the local stability condition Eq. 17. Therefore, the convex blending with the membership functions hi(z̲) is obsolete and the form of the membership functions has no influence on the design, as just the i th submodel is included in the observer design—see Section 2.3.)

The augmented state, input and output matrices Ãi,B̃i and C̃, as well as the augmented steady system states x̲̃c,i and steady input states u̲c,i, used within this contribution, are given in Table A2 to Table A6. Additionally, the state feedback gain matrices Ki and error-feedback gain matrices LiBw are listed in Table A7 and Table A8.

2.4.4 Decomposed Wind Turbine System and Subsequent Lyapunov Stability Analysis

Mechanical and mechatronic systems, like wind turbines (WT), are characterised by the pole locations of the coupled mechanical components. Due to the widely spaced, open-loop pole locations of the WT system (see Figure 4 and Section 4), it is advantageous for the controller and observer design models to decompose the coupled, nonlinear system submodel (Ai, Bi) into decoupled reduced submodels—denoted with component models—just comprising the dynamics of the individual, mechanical main components like the rotor blade submodels (AiB,BiB) and drive train submodels (AiR,BiR) (also described in (Pöschke et al., 2020), but now upgraded with the rotor blade submodels (AiB,BiB)).

FIGURE 4
www.frontiersin.org

FIGURE 4. Pole locations of the error dynamics (achieved within the wind speed observer design) for an increasing upper bound αB,max (from left to right), applied to the global Lyapunov approach (upper row) and local Lyapunov approach (lower row), restricted to the four submodels i ∈[15,18] that are relevant for the wind turbine simulations with the prescribed wind speeds v(t) between 14 m/s and 16 m/s. The Open-Loop, real-valued poles sP,iOL,p (with p = 1) of the wind model and the open-loop, conjugate complex-valued poles of the blade model sP,iOL,p (with p = 2 ∨ p = 3) are depicted in grey colour, while the closed-loop real-valued (p = 1, see Section 3.2 and Table 2) and closed-loop, conjugate complex-valued (p = 2 ∨ p = 3) poles sP,iw,p of the global wind observers(withw[A,E]) and the local wind observers(withw[F,J]) are depicted in green or orange colour and are distinguished by the superscripted index w.

Thereby, couplings between the components’ degree of freedom are neglected within the component controller and observer design models, like the missing axial coupling between drive train and main frame/tower depicted in Figure 2 (just plain ball bearings are defined). Despite this substantial assumption, the component design models achieve satisfying controller performance, while increasing the controller design flexibility significantly.

The corresponding design models are derived from the elaborated NREL FAST 5 MW reference wind turbine (Jonkman, 2016; Jonkman and Buhl (2005) and Jonkman et al. (2009)) simulation model, based on a convex sum of 27 linearised submodels (defined for piecewise equidistant, steady and effective wind speeds from v ∈ [3 m/s, 25 m/s]; see Table A1), received from the NREL FAST WT linearisation (Jonkman and Jonkman, 2016). Within this linearisation, all aerodynamic and structural dynamic characteristics defined in the elaborated NREL FAST 5 MW reference wind turbine are accounted (see also explanations given in (Pöschke et al., 2020)).

Once the component controllers and observers are designed, the corresponding state-feedback gains (KiR, KiB) and observer gains LiR, LiB) are superposed for the overall controller gain Ki and observer gain Li. The stability of the convexly blended i submodels is analysed with the superposed gains Ki and Li in a subsequent, global Lyapunov stability analysis (see Section 2.3.5), separately for the controller and error dynamics (Yoneyama et al. (1998), Ma et al. (1998)). Although, the separation principle and overall stability proof, respectively of separately designed observer and controller is just valid for measurable premise variables z̲ (Yoneyama et al. (1998), Ma et al. (1998)), it is shown in (Pöschke et al., 2022) for a separated observer and controller design, that the overall stability is also ensured for reconstructed premise variables ẑ̲, under the assumption of a maximum estimation error in the premise variable with ẑ̲z̲. Therefore, the stability proof and separation principle, respectively holds for ẑ̲z̲. Even though the maximum error estimation is missing in this contribution, it is assumed, that ẑ̲z̲ and the separation principle holds. The missing error estimation and overall stability proof corresponding to (Pöschke et al., 2022) has to be given in future work. That is, within this contribution just separated controller and observer syntheses are executed in regard to a pragmatic design process that intends for less conservative state and error state feedback gains Ki and Li, like explained in Section 1. Thus, the design process is focused on an automated controller and observer design, i.e., the Lyapunov approach was selected to achieve the controller and observer gains in a systematic and traceable procedure, rather than to ensure the overall stability of the observer-based system dynamics. Though, the stability condition of the Lyapunov approach is exploited (separately in the controller and observer design) to abbreviate the iterative design process, i. e., all controller and observer design parameter specifications, resulting in unstable dynamics, are eliminated within the controller and observer synthesis to expedite the design process.

2.5 LMI Constraints of the Pole Regions

With the linear matrix inequalities Eqs 13, 16, 19 and Eq 21 the pole locations are just restricted to the left half of the complex pole map. Additional constraints need to be defined to tighten the pole location on smaller pole regions. As described in (Chilali and Pascal, 1996), additional bounds specified in the complex pole map can be transformed into LMI, which are applied to wind turbine control e.g., in (Pöschke et al., 2019). Within this contribution, just vertical upper and lower bounds (representing the maximum and minimum decay rate αB,max and αB,min) and a cone angle θ (representing the Damping ratio D with D = cos(θ)) are used and described in the Appendix Section A1.3. With these constraints, pole regions with symmetrical trapezium shape are defined (see Figure 4).

3 Simulations and Results

With the local Lyapunov approach, the general objective of a more flexible specification of the pole locations and increased consistency in the desired and achieved closed-loop system dynamics is intended (see Section 2.3.4). This general objective is analysed within this contribution for a wind turbine specific, particular objective—the decreased observer performance and feedforward actuation—described in Section 3.1. The achieved results from wind turbine simulations are presented in Section 3.2.

3.1 Simulation Design: Particular Objective of the Local Lyapunov Approach Within this Contribution

Within this contribution, the general objective of the local Lyapunov approach (see Section 2.3.4) is assessed for a particular wind speed observer performance, hereinafter denoted with the particular objective, supposed to be beneficial regarding load mitigation: For turbulent wind excitation, high mechanical loads (especially fatigue loads) often result from brisk feedforward actuation5. Therefore, the proposed local Lyapunov approach shall yield for a decreased disturbance reconstruction v̂ performance—affecting the premise variable v̂—to attenuate the premise variable v̂ driven feedforward actuation u̲FF(hi(v̂)) (see Eq. 5) with z̲v̂) and to increase the feedback compensation u̲FB(hi(v̂)) (see Section 2.1.2). That is for this particular objective, the closed-loop dynamics is defined in such a way, that the performance of the wind speed v̂ reconstruction and the precision of the feedforward actuation u̲FF(hi(v̂)) is intentionally lowered to achieve rising deviations, which are compensated by increased feedback controller actuation u̲FB(hi(v̂)). As the error-feedback gains Li govern the observer performance and these gains are achieved from the observer design according to the specified error-feedback pole regions, the pole region specification is varied by shrinking its size. That is, a significant modification of the pole regions is conducted with the local Lyapunov approach to achieve the particular objective of a decreased observer performance, resulting in mitigated mechanical loads, especially mitigated fatigue loads. To lower the wind speed observer performance and error dynamics (i.e., the wind speed v̂ depending error-feedback gains Li in iNrhi(v̂)LiCx̲x̲̂=L(v̂)Ce̲6), the upper bound αB,max of the error dynamics’ pole region is increased, i.e., shifted towards the imaginary axis in the left half of the complex pole map, while the lower bound αB,min (located close to the open-loop poles) is kept constant (see Figure 4). It is expected, that the error-feedback gains Li decrease with increasing upper bounds αB,max (αB,maxLi), as the distances between open-loop poles and closed-loop poles of the error dynamics ΔsP,iw,p(αB,max) decrease with increasing upper bounds αB,max and the quantity of the error-feedback gain depends on this distance |Li|=f(ΔsP,iw,p)7.

To distinguish the pole locations sPw (see Figure 4), error-feedback gains Liw (see Table A8) or mechanical loads like Seqw (see Figure 8), resulting from a number of w different error-feedback pole region specifications, the superscript index w (denoting the w different error-feedback pole regions and corresponding wind speed observers) is introduced in the following. To analyse the error-feedback gains LiBw of the Blade model-based wind speed observer, the following state space equation is relevant (see Eq. 7), describing the reconstructed blade component dynamics:

x̂̇Bx̂̈Bv̂̇x̲̃̂̇=iNrhiz̲Aix̲̃̂x̲̃c,i+Biu̲u̲c,i+iNrhiv̂LiBw,1LiBw,2LiBw,3LiBw100CBxBx̂BẋBx̂̇Bvv̂e̲
=iNrhiz̲Aix̲̃̂x̲̃c,i+Biu̲u̲c,i+iNrhiv̂LiBw,100LiBw,200LiBw,300xBx̂BẋBx̂̇Bvv̂(22)

with the j components x̃̂̇j of the reconstructed and augmented state vector x̲̃̂̇ and with j matrix elements LiBw,j of the error-feedback gain LiBw.

Within this contribution, the general objective of an increased flexibility in the observer-based controller design and increased consistency of the desired and achieved system dynamics (see Section 2.3.4) is analysed with the particular objective of an decreased observer performance and feedforward actuation, resulting from shrinking pole regions. That is, for both Lyapunov approaches identical upper bound variation and shrinking pole regions are defined and the flexibility and consistency of both approaches are compared with the help of several metrics (like pole locations, error-feedback gains, reconstructed wind speeds, pitch angles, pitch rate and rotation speed deviations): Regarding the flexibility, the minimum pole region dimension is determined. That is, for all poles it is examined, if the poles are located inside the imposed pole region after executing the error-feedback gain Li design. The intended, particular objective of a decreased observer performance of the local Lyapunov approach is fulfilled, if this approach leads to a smaller, minimum pole region than the global Lyapunov approach, enabling the specification of tighter performance constraints for the system dynamics. Regarding the consistency of desired and achieved system dynamics, it is analysed, if the shrinking poles regions result in decreased feedforward pitch angles βFF and increased feedback pitch angles βFB with decreased pitch rates β̇ and increased, resulting rotation speed deviations Δω.

In addition, the control objectives of mitigated ultimate and fatigue loads are analysed for identical WT controller pole region specifications, but related to both disturbance observer design approaches.

3.2 Simulation Results: Disturbance Observer Variation, Resulting System Dynamics and Mechanical Loads

For each Lyapunov approach, five different disturbance observers for the global Lyapunov approach (denoted in the following with the superscripted index winA,E) and for the local Lyapunov approach (denoted with wF,J are designed—based on five different pole regions with shrinking size—and the resulting closed-loop dynamics as well as the resulting mechanical loads on the WT components are analysed.

Pole regions and submodels

In Table 1, the pole region specifications for both design approaches are listed. For the analysed wind time series excitation with prescribed wind speeds v(t) between 14 m/s and 16 m/s, just four of the i ∈ [1,27], from linearisation achieved submodels (denoted with the subscripted index i) are convexly blended. Therefore, just these four submodels (Ai, Bi) with i ∈ [15,18] are taken into account in the results and discussion.

TABLE 1
www.frontiersin.org

TABLE 1. Description of the TS disturbance observers (i.e., wind speed observers) (for the wind speed observers w in A to E, based on the global Lyapunov-approach and for the wind observers w in F to J, based on the local Lyapunov-approach, with the cone angle, and the lower and the upper bounds of the five pole regions (see Figure 4), defined by the Blade error state damping DB and the Blade decay rates αB,min and αB,max).

Pole locations

Figure 4 shows the resulting p pole locations sP,iOL/w,p of the corresponding open-loop dynamics (denoted with the superscripted index OL), as well as the closed-loop dynamics (denoted with the superscripted index w for the w inA,E) globalwind speed observers and wF,Jlocal wind observers) and error dynamics, respectively, achieved within the disturbance observer design, in the complex pole map. As the disturbance observer design model and wind speed observer design model, respectively, consists of the vibrating Blade model (with its two states x̂B,x̂̇B see Eq. 22) and the non-vibrating wind model (with the wind speed state v̂), the design model posses three poles (i.e., p ∈ [1,3]): the conjugated-complex blade poles sP,iOL/w,23 (inside the pole map, for p = 2 ∨ p = 3) and the real-valued wind model pole sP,iOL/w,1 (on the real axis of the pole map, for p = 1). In Table 2 the distances

ΔsP,iw,p=ResP,iOL,pResP,iw,p2+ImsP,iOL,pImsP,iw,p2(23)

between open-loop poles sP,iOL,p and closed-loop poles sP,iw,p in the pole map are given for each of the p ∈ [1,3] poles of the wind and blade model. Additionally, the related average distance

ΔsP,iw,p̄=13p=13ΔsP,iw,p=13ΔsP,iw,1+ΔsP,iw,2+ΔsP,iw,3(24)

of the single submodel i and the average distance

ΔsP,īw,p̄=14i=1518ΔsP,iw,p̄=14ΔsP,15w,p̄+ΔsP,16w,p̄+ΔsP,17w,p̄+ΔsP,18w,p̄(25)

of all four submodels are listed in Table 2.

TABLE 2
www.frontiersin.org

TABLE 2. Distances ΔsP,iw,p between the open-loop pole location sP,iOL,p and closed-loop pole locations sP,iw,p of each of the p ∈ [1,3] poles (with p = 1 for the real-valued pole of the wind model and p = 2,3 for the complex-valued poles of the blade model) of a single submodel i in the pole map for the wth wind speed observer; average distance ΔsP,iw,p̄ of the single submodel i; average distance ΔsP,īw,p̄ of all four submodels.

Error feedback gains

The related error-feedback gain matrices LiBw,j (with their j elements LiBw,j described in Eq. 22) are summarised in Table A8. To assess the LiBw,j deviations resulting from the shrinking pole regions, two different mean Euclidian norms LīBw,j̄2 and LīBw,j̄3/j̄1,22 are calculated as metrics of the error-feedback gains Li,Bw,j for each of the wA,Eglobal wind speedobservers and wF,Jlocal wind speed observers and both metrics averaging all, four incorporated submodels i ∈ [15, 18]:

The metric LīBw,j̄2 cumulates all three LiBw,j elements (with j ∈ [1, 3])

LīBw,j̄2=14i=1518LiBw,j̄2=14L15Bw,j̄2+L16Bw,j̄2+L17Bw,j̄2+L18Bw,j̄2(26)

withLiBw,j̄2=LiBw,12+LiBw,22+LiBw,32. The metrics LīBw,j̄3/j̄1,22 are calculated separately for LiBw,j̄3 (to gain the wind speed state v̂̇, see Eq. 22) and LiBw,j̄1,2 (to gain the blade states x̂̇B and x̂̈B):

LīBw,j̄3/j̄1,22=14i=1518LiBw,j̄3/j̄1,22=14L15Bw,j̄3/j̄1,22+L16Bw,j̄3/j̄1,22+L17Bw,j̄3/j̄1,22+L18Bw,j̄3/j̄1,22withLiBw,j̄32=LiBw,32andLiBw,j̄1,22=LiBw,12+LiBw,22.(27)

The results are depicted in Table 3.

TABLE 3
www.frontiersin.org

TABLE 3. Mean Euclidean norm LiBw,j̄2 and average, mean Euclidean norm LīBw,j̄2 of the error-feedback gain matrices Li(vi) of the global wind speed observers A to E (i.e., w ∈ [A, E]) and local wind speed observersFtoJ (i.e., w ∈ [F, J], see Table A8) for increasing upper bounds αB,max (with LīBw,3̄2 for the average, mean Euclidean norm of the wind model error-feedback gains (j = 3) and LīBw,(1,2)̄2 for the average, mean Euclidean norm of the Blade error-feedback gains (j = 1,2))

Membership functions

To visualise the convex blending of direct adjacent submodels (Ai, Bi), the time series of the membership functions hi for a step-shaped wind time series are depicted in Figure 5. Note: As the premise variable in this contribution is defined by one parameter (z̲v̂), the membership function hi(z) is identical to the weighting function wk,l(z) (see Section 2.4.3, especially footnote4). Therefore, just two submodels are blended at the same time (for all operating points between two steady state operating points OPi; see also the triangular-shaped hi(z) progression (with hi(z) ≡ wk,l (zk) for zk = z1 = z with kmax = 1) in Figure 1).

FIGURE 5
www.frontiersin.org

FIGURE 5. Exemplarily time series of the membership functions hi for the global wind speed observerA (with i ∈[15,18]) for the step-shaped wind time series depicted in Figure 6.

Wind turbine simulations, actuation signals, pitch rate deviations, rotation speed deviations and resulting mechanical loads

To assess the closed-loop system dynamics (see Figure 6), a disturbing step-shaped time series of 150 s duration is applied to the system, visualising the effect of the particular wind speed observer design (with decreased performance, see Section 3.1) on the actuation signals. The resulting time series v̂w(t), βFFw(t) and βFBw(t) were achieved with NREL FAST 5 MW reference wind turbine simulations, embedded in a Matlab Simulink model, using the wind speed observer-based PDC controller Eq. 5 (i.e., for all simulations the same PDC feedback-controller8 is utilised) with step-shaped wind time series as disturbing excitation. For clarity, the simulations are restricted to the full load range with pitch angle actuation, i.e., the generator torque is kept constant (see Section 2.4.2 and Table A1).

FIGURE 6
www.frontiersin.org

FIGURE 6. Time series segment of wind speed reconstruction v̂ (left column), feedforward pitch actuation βFF (mid column) and feedback pitch actuation βFB (right column), with the observer-based controller and step disturbance signals for an increasing upper bound αB,max resulting in shrinking pole regions, applied to the global Lyapunov approach (with w ∈ [A, E], see upper subplots) and local Lyapunov approach (with w ∈ [F, J], see lower subplots).

The given wind speeds v and the reconstructed wind speeds v̂w of the step-shaped wind time series are depicted in the left column of Figure 6, the actuated feedforward and feedback pitch angles βFFw(t) and βFBw(t) are depicted in the mid and right column of Figure 6. The resulting ultimate and mean pitch angle deviations, i.e. the pitch rates max(β̇) and mean(β̇) are depicted in the left column of Figures 7A,B; the closed-loop system dynamics, i.e. the ultimate and standard drive train rotation speed deviations max(Δω)=max(ωωrωr) and std(Δω)=std(ωωrωr) (with the rated rotation speed ωr), are depicted in the right column of Figures 7A,B.

FIGURE 7
www.frontiersin.org

FIGURE 7. Actuation speed and pitch angle deviations, respectively (so called pitch rate β̇=dβ/dt), i.e. ultimate max(β̇w) and mean mean(β̇w) pitch rates for the w[A,E]global wind speed observers and w[F,J]local wind speed observers (normalised to max(β̇A) and mean(β̇A); see left column of subplot A,B). Rotation speed deviations Δωw (with Δωw=ωwωrωr and the rated rotation speed ωr), i.e. ultimate max(Δωw) and standard std(Δωw) rotation speed deviations [normalised to max(ΔωA) and std(ΔωA); see right column of subplot A,B]. The ultimate deviations max(β̇w) and max(Δωw) (see subplot A) result from the ultimate load analysis and the mean deviations mean(β̇w) and standard deviations std(Δωw) (see subplot B) result from the fatigue load analyses of the step-shaped wind time series, depicted in Figure 8.

In addition, the step-shaped wind time series – representing a typical wind increase and wind decrease event (i.e. a disturbing step up and step down signal) within a turbulent wind excitation time series – is used to assess the ultimate and fatigue loads resulting from the closed-loop system dynamics of both approaches. For this load assessment, five different loads are analysed that are crucial or at least very important for the wind turbine’s main component design of tower, blades and the drive train:

• The tower bending moment in wind direction (also denoted with the fore-aft bending moment, TwrBsMyt) and normal to the wind direction (also denoted with the side-to-side bending moment, TwrBsMxt), calculated for the tower base,

• the blade bending moment in wind direction (i.e., resulting from blade bending out of the rotor plane, also denoted with the flapwise or out-of-plane bending moment, RootMyb1) and blade bending moment inside the rotor plane (edgewise or in-plane bending moment, RootMxb1), calculated for the blade root (next to the hub body) of the first blade, and

• the torsional torque (ΔTq) along the drive train axis.

The ultimate and fatigue loads, resulting from the closed-loop system dynamics, are depicted in Figure 8. To eliminate effects, resulting from simulation initialisation, the time period 0stCutOff ≤ 10s of all time series is not taken into account in the load evaluation. (Note: If the fatigue loads in Figure 8 just differ even in the third or fourth decimal place, these differences are of significance. Because, the listed fatigue loads in Figure 8 result from a wind times series of extremely short duration (of 150 s), while wind turbines typically operate for 20 years with approximately 97% availability. That is, also fourth decimal place differences in the fatigue loads, depicted in Figure 8, do have an enormous impact on the expected wind turbine lifetime, as all damages cumulate over the complete lifetime. This cumulation does not apply to ultimate loads, as these loads occur rarely in a wind turbine’s lifetime).
FIGURE 8
www.frontiersin.org

FIGURE 8. Ultimate loads maxw (…) and fatigue loads Seqw() (also denoted with Damage Equivalent Loads or Damage Equivalent Amplitudes Seq) resulting from the closed-loop dynamics of a step-shaped wind time series with a mean wind speed of v̄=14m/s (see Figure 6), for the w different global wind speed observers (resulting from the global Lyapunov design approach withwA,E) and the wlocal wind speed observers(resulting from the local Lyapunov design approach withw[F,J]), normalised to the ultimate load maxA (…) or fatigue load SeqA() of the global wind speed observer A. For the ultimate and fatigue loads, the tower bending moment TwrBsMyt and TwrBsMxt (at the tower base—top row), the blade bending moment RootMyb1 and RootMxb1 (at the blade root section—mid row) and the drive train torque ΔTq (bottom row) are depicted.

4 Discussion

Both Lyapunov approaches fulfill the particular objective to decrease the disturbance observer performance and feedforward actuation, while increasing the feedback actuation for shrinking pole region dimensions, resulting in mitigated mechanical loads (see Section 3.1). But the local approach achieves significantly increased flexiblity and consistency of the desired and resulting system dynamics than the global approach:

Pole locations

The open-loop poles of the wind turbine system, depicted with gray symbols in Figure 4, are widely spaced, as significant distances between the real-valued wind model poles sP,iOL,1 (on the real axis of the pole map) and the conjugated-complex blade poles sP,iOL,23 (in the pole map) exist. With increasing upper bounds αB,max, resulting in shrinking pole regions9, the closed-loop pole locations sP,iw,p are shifted for both design approaches inside the left half of the pole map in direction of the imaginary axis and towards the open-loop poles sP,iOL,p. Thereby, the average distances ΔsP,īw,p̄ in the complex pole map decrease (i.e., ΔsP,īw,p̄>ΔsP,ī(w+1),p̄, e.g., ΔsP,īB,p̄>ΔsP,īC,p̄, see Figure 4 and Table 2). Just for the last global and local wind speed observers the distances increase, i.e., ΔsP,īD,p̄<ΔsP,īE,p̄ and ΔsP,īI,p̄<ΔsP,īJ,p̄, as the closed-loop poles (and their real-values Re(sP,iw,p), respectively) are shifted beyond the open-loop poles Re(sP,iOL,p) in direction of the imaginary axis. Therefore, those two wind speed observers E and J are not accounted in the following evaluations, but they are relevant for the local approach assessment, as described in the following with regard to the pole region violation.—For the wind speed observer poles, based on the local Lyapunov approach (see w ∈ [F, I] in Table 2), the pole distances ΔsP,īw,p̄ decrease steadily, while the distances ΔsP,īw,p̄ for the wind speed observer poles, based on the global Lyapunov approach (see w ∈ [A, D]), show a local maximum for the wind speed observer B (i.e., ΔsP,īB,p̄>ΔsP,īA,p̄>ΔsP,īC,p̄>ΔsP,īD,p̄). Thus, the local Lyapunov approach is rated to be more coincident in regard to a desired pole location specification. Additionally, it is obvious from the pole locations depicted in Figure 4, that the local Lyapunov approach is more effective and flexible, respectively, as all poles are located inside the specified pole regions, while the global Lyapunov approach is not able to satisfy the pole region specifications for the smallest pole region, resulting in poles located outside the specified pole regions (see the pole location sP,iE,p of the global wind speed observer E in Figure 4E). That is, the local Lyapunov approach is more flexible in assigning desired pole locations than the global Lyapunov approach.

(Note, that the LMI solver is capable to find solutions, even if the pole region LMIs are violated. However, the resulting poles are still located in the left half of the complex pole map and fulfill the basic stability condition for the corresponding closed-loop or error dynamics.)

Error-feedback gains

With the decreasing pole distances ΔsP,īw,p̄, also the error-feedback gains LiBw,j—representing a measure or the effort of all error states acting on the wth observer—are mitigated. For all10 local wind speed observer designs these gains decrease expectedly with increasing upper bound αB,max and with the reduced pole distances ΔsP,īw,p̄ (i.e., LīBF,j̄2>LīBG,j̄2>LīBH,j̄2>LīBI,j̄2). Corresponding to the pole distances ΔsP,īw,p̄, the local wind speed observer gains LiBw,j are mitigated steadily (and yield lower gains LīBB,j̄2>LīBG,j̄2,LīBC,j̄2>LīBH,j̄2andLīBD,j̄2>LīBI,j̄2), while the global wind speed observer gains show a local maximum of LīBw,j̄2 for the wind speed observer B (i.e., LīBB,j̄2>LīBA,j̄2>LīBC,j̄2>LīBD,j̄2), demonstrating the increased consistency of the desired and achieved error-feedback gains LiBw,j of the local Lyapunov approach.

Wind speed reconstruction and actuation signals

With the mitigated error-feedback gains LiBw,j, the reconstructed states x̲̂, especially the reconstructed wind speeds v̂, are mitigated, too (see Eq. 711): While the reconstructed wind speed v̂w(t1) of a single and arbitrary time point t = t1 decreases steadily for the wind speed observer design with local Lyapunov approach (i.e., v̂F(t1)v̂G(t1)>v̂H(t1)>v̂I(t1)[>v̂J(t1)]10, see left column in Figure 6), the reconstructed wind speed v̂w(t1) for the wind speed observer design with global Lyapunov approach decreases unsteadily (i.e., v̂A(t1)>v̂D(t1)[>v̂E(t1)]>v̂C(t1)>v̂B(t1), corresponding to the unsteady decrease of the mean Euclidean norm of the wind error state gains LīBw,3̄2 of the global wind speed observers (with w [A, E], see Table 3; i.e., LīBA,3̄2>LīBD,3̄2>LīBC,3̄2>LīBB,3̄2)10.

The mitigated, reconstructed wind speeds v̂w of both approaches result in the following actuation signals: The mitigated wind speeds v̂w lead to less dominant feedforward pitch actuation βFF—just governed by the wind speed driven feedforward actuation u̲FF(hj(z̲)) (with uFFβFF and z̲v̂w) according to Eq. 5 and the steady state pitch angles βc,i (listed in Table A1)—and increase the feedback pitch angles βFB (see mid and right column in Figure 6). That is, corresponding to the decreased, reconstructed wind speeds v̂w (and mean wind state feedback gains LīBw,3̄2), the feedforward pitch angles of the local wind speed observers βFFw (with w ∈ [F, I]10) are mitigated steadily (i.e., βFFFβFFG>βFFH>βFFI[>βFFJ], see mid column in Figure 6) and yield steadily increased feedback pitch angles (i.e., βFBF<<βFFI[<βFBJ], see right column in Figure 6), while the feedforward pitch angles of the global wind speed observers βFFw (with w ∈ [A, D]) decrease unsteadily (i.e., βFFA<βFFD<βFFC<βFFB) and yield correspondingly increasing feedback pitch angles βFBw (with w ∈ [A, D]). Thus, the intended particular objective of a steadily decreasing feedforward-actuation with increasing feedback-compensation (see Section 3.1) is achieved with the local, not with the global Lyapunov wind speed observer design approach.

Pitch rate deviations and rotation speed deviations

The pitch angle deviations result in corresponding pitch and rotation speed metrics: With the steadily mitigated premise variable v̂F>v̂G>v̂H>v̂I of the local wind speed observer F to I, the ultimate pitch rates max(β̇)(=max(β̇FF+β̇FB)withmax(β̇F)max(β̇G)>max(β̇H)>max(β̇I)) and mean pitch rates mean(β̇) (with mean(β̇F)mean(β̇G)>mean(β̇H)>mean(β̇I)) decrease steadily, too (see left column in Figures 7A,B), while the unsteadily decreasing wind speeds v̂A>v̂D>v̂C>v̂B of the global wind speed observers A to D lead to unsteadily decreasing ultimate pitch rates max(β̇A)>max(β̇D)>max(β̇C)>max(β̇B) and mean pitch rates mean(β̇A)>mean(β̇D)>mean(β̇C)>mean(β̇B). As the pitch rates β̇ decrease, increasing system dynamic deviations appear, e.g. represented by the metric of the drive train rotation speed deviations max(Δω) and std(Δω), that increase steadily for the local wind speed observers (with max(ΔωF) ≈ max(ΔωG) and std(ΔωF) ≈ std(ΔωG)) and unsteadily for the global wind speed observers (see right column in Figure 7 A,B). Thus, the local Lyapunov approach achieves the expected system dynamics with increased consistency between the desired and achieved pitch rate deviations and rotation speed deviations, compared to the global Lyapunov approach.

Load mitigation

The ultimate and fatigue loads resulting from closed-loop dynamics with the step-shaped wind excitation (depicted in Figure 8) are satisfying for both approaches, but especially for the local Lyapunov approach. In the following, the ultimate and fatigue loads are first evaluated separately for each of the two Lyapunov approaches (see Table A9), then the loads from both approaches are compared with each other (see Table A10).

• Ultimate Loads (see Figure 8A): The ultimate tower bending moments max(TwrBsMyt) and max(TwrBsMxt), as well as the ultimate in-plane blade bending moments max(RootMxb1) are mitigated for both Lyapunov approaches, and for the local Lyapunov approach these ultimate loads decrease even steadily (see lines 1 to 3 in Table A9; with one marginal exception for maxF(TwrBsMxt) < maxG(TwrBsMxt)). The ultimate out-of-plane blade bending moments max(RootMyb1) and the ultimate drive train torque max(ΔTq) increase slightly, but not significantly (see Figure 8A and lines 4 to 5 in Table A9). – Comparing both approaches with each other, for most pole regions lower ultimate loads are achieved with the local Lyapunov design approach (see Figure 8B and lines 1 to 5 in Table A10).

  If a controller variation yields comparable or even lower ultimate loads, the controller (design approach) assessment depends on the fatigue loads.

• Fatigue Loads (see Figure 8B): The fatigue tower bending moments Seq(TwrBsMyt) and Seq(TwrBsMxt), as well as the fatigue drive train torque SeqT) are mitigated for both Lyapunov approaches, and for the local Lyapunov approach these fatigue loads decrease steadily (see lines 6 to 8 in Table A9; with one exception for SeqF(TwrBsMxt)<SeqG(TwrBsMxt)). The fatigue out-of-plane blade bending moments Seq(RootMyb1) and in-plane blade bending moments Seq(RootMxb1) increase slightly, but for the local Lyapunov approach the out-of-plane blade bending fatigue loads Seqw(RootMyb1) decrease steadily (see Figure 8B and lines 9 to 10 in Table A9). – Comparing both approaches with each other, for most pole regions lower fatigue loads are achieved with the local Lyapunov design approach, again (see Figure 8B and lines 6 to 10 in Table A10).

  Additionally, the local wind speed observer design approach with the smallest pole region (see wind speed observer J) achieves the lowest fatigue loads for those bending moments and torsional torque compared to the fatigue loads, resulting from the global wind speed observer design (compare the fatigue loads of local wind speed observer J with the fatigue loads of all global wind speed observers for Seq(TwrBsMyt), Seq(TwrBsMxt)12, Seq(RootMyb1) and SeqTq) in Figure 8). – Just the in-plane blade fatigue bending moments Seq(RootMxb1) increase for the local wind speed observer design approach (as well as for the global approach). This fatigue load increase might result from the increasing rotation speed deviations max(Δω) and std(Δω). Because, for identical wind times series and system excitations, respectively, increasing fatigue loads result from increasing amplitude and/ or increasing load cycles. As the rotation speed deviations max(Δω) and std(Δω) increase with the shrinking pole region dimension (see max(Δω) and std(Δω) for WindObs w with w ∈ [G, J] in the right column of Figures 7A,B), it seems reasonable, that the fatigue load increase results from the increasing rotation speed, rather than from an amplitude increase. This assumption is also confirmed by the ultimate loads that decrease with the shrinking pole region size (see max(RootMxb1) for WindObs w with w ∈ [F, J] in Figure 8), i.e. not the amplitudes of ultimate and fatigue loads, but the load cycles of the fatigue loads increase. Additionally, with the increasing rotor speed deviations, the number of blade passages through the tower shadow13 increases, resulting in increasing load cycles and leading to increased fatigue loads.

Therefore, it seems promising as a subject of future work, to split the wind speed observer, based on a local Lyapunov approach, into two, separated wind speed observers: A feedforward wind speed observer—specified with a similar pole region (like wind speed observer I, used within this contribution)—with decreased reconstruction performance to assign the premise variable v̂FF to the feedforward actuation u̲FF(h(v̂FF)). To compensate the decreased feedforward pitch rates β̇FF, an additional feedback wind speed observer with increased performance will be implemented, to assign its reconstructed wind speed v̂FB to the feedback actuation. For this feedback wind speed observer, a tight pole region will be specified, located left from the open-loop poles, resulting in significantly increased error-feedback gains Li,FB and reconstructed wind speeds v̂FB, so that the feedback pitch rates β̇FB increase, resulting in mitigated rotation speed deviations and fatigue loads. Summing up the load evaluation, the local Lyapunov approach is rated to gain an increased consistency between intended and achieved load mitigation compared to the global approach. While the ultimate loads, achieved with the global and local wind speed observer approach are comparable, the local approach achieves an increased fatigue load mitigation. Because, for steadily shrinking pole regions with steadily decreasing feedforward actuation, the fatigue loads decrease steadily, too.

5 Conclusion

With this contribution a local Lyapunov approach is introduced for controller and observer design in the Takagi-Sugeno framework. Compared to the common global Lyapunov approach, a more dedicated closed-loop dynamic is intended with the local Lyapunov approach, i.e., an increased flexibility in the design process and increased consistency between desired and achieved system dynamics is aspired.

The applicability and effectiveness of the local Lyapunov approach to wind turbine control is analysed with wind turbine simulations. The simulation results show, that the local Lyapunov approach makes it possible to influence the pole locations, the resulting error-feedback gains and closed-loop system dynamics more flexible and with an increased consistency between desired and achieved system dynamics than the global Lypunov approach. That is, with the local Lyapunov approach, the assignment of smaller pole regions is possible, enabling a higher flexibility in assigning desired system dynamics. Additionally, the local Lyapunov approach reaches an improved similarity between the desired and achieved closed-loop system dynamics, indicating the increased consistency of the local Lyapunov approach. As the local Lyapunov approach is less conservative, but more dedicated to the desired system dynamics, i.e., the design process achieves an increased flexibility and increased consistency, it is rated to have a higher potential in fulfilling primary and secondary control objectives, like energy yield optimisation and load mitigation in wind turbine application.

The evaluation of the local Lyapunov approach will be continued in oncoming studies, with the controller structure adapted to the new possibilities arising from the local design approach, like the separated reconstruction of the feedforward and the feedback premise variable in a split wind speed observer.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

Eckhard Gauterin and Horst Schulte devised the basic Takagi-Sugeno observer-based feedforward control concept for wind turbine control and conceptualized this study. Florian Pöschke developed the linearised wind turbine model in Takagi-Sugeno framework and conceived the local Lyapunov approach. Florian Pöschke expedited the implementation and development of the control concept in the simulation software, while Eckhard Gauterin conducted the simulation studies. All three authors analysed the results and Eckhard Gauterin created the manuscript, that was reviewed by all three authors.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

1With individual input matrices Bi a weighted combination of the ith submodel with all and the direct adjacent submodels (see Section 2.1.1), respectively, is derived for the closed-loop dynamics, due to the individual weighting ihi(z̲)Bi of the individual input matrices Bi for each submodel. Therefore, the double summation i j is necessary in Eq. 6.

2With a common output matrix CiC the weighted combination of the ith submodel with all and the direct adjacent submodels, respectively, is superfluous for the reconstructed closed-loop dynamics, as ihi(z̲)C=Cihi(z̲)=C1=C holds for all submodel. Therefore, the single summation i is sufficient in Eq. 7 and Eq. 9.

3Note: Within this contribution just the blade tip translations xB,1/2/3 are assumed to be measurable, whereat the mean value xB=(1/3)(xB,1+xB,2+xB,3) is utilised for calculating the error eB=xBx̂B, fed back and amplified with the error gains Li to reconstruct the blade tip speed x̂̇B, blade tip acceleration x̂̈B and temporal wind speed variations v̂̇B.

4The triangular shape of the membership functions holds, if just one premise variable zk = z (with k = kmax = 1) is defined. Because in this case, the membership function hi(z) (Eq. 1) is identical to the triangular-shaped weighting function wk,l(z), see Eq. 2.

5These brisk feedforward-actuation result from feedforward specifications without dynamics (i.e., without any damping influence, e.g., from the Ki gains), but simple convex blending of the steady states u̲c,j (see u̲FF in Eq. 5).

6with iNrhi(v̂)LiCx̲x̲̂=iNrhi(v̂)Li=:L(v̂)Ce̲=L(v̂)Ce̲, see Eq. 7 (with z̲v̂) and Eq. 8.

7As the pole location distance ΔsP,iw,p depends on the difference between sP,iOL,p and sP,iw,p (i.e., ΔsP,iw,p=f(sP,iOL,psP,iw,p), see Section 3.2 with sP,iOL,p=eig(Ai) and sP,iw,p=eig(AiLiwC), i.e., ΔsP,iw,p=f(eig(Ai)eig(AiLiwC))), this distance ΔsP,iw,p in-/decreases, if the error-feedback gain Liw in-/decreases—and vice versa, i.e., Liw in-/decreases for in-/decreasing ΔsP,iw,p.

8PDC feedback controller: ct1010001

9to realise the shrinking pole region size, compare the pole regions’ size (and location) in Figure 4 for the wind speed observer design A and F with E and J

10The global and local wind speed observers E and J are not taken into account, because of their (closed-loop) pole locations, which are moved beyond the open-loop pole locations, as explained before in the subsection Pole locations.

11If Eq. 7 is evaluated for a single and arbitrary time point, it is obvious, that x̲̂̇ (and x̲̂) decreases, if LiBw,j is mitigated, while all other parameters and states do not change.

12with two exceptions for the tower side-to-side-bending moments SeqB(TwrBsMxt)<SeqJ(TwrBsMxt) and SeqC(TwrBsMxt)<SeqJ(TwrBsMxt) (see Figure 8B and line 7 in Table A9 as well as line 7 in Table A10.

13Tower shadow (effect): The tower poses as an obstacle in the inflow that increases the dynamic pressure and decreases the wind speed in front, i.e. in upwind direction of the tower.

14with two exceptions for the tower side-to-side-bending moments SeqB(TwrBsMxt)<SeqC(TwrBsMxt)<SeqJ(TwrBsMxt)

15Tower shadow (effect): The tower poses as an obstacle in the inflow that increases the dynamic pressure and decreases the wind speed in front, i.e., in upwind direction of the tower

References

Bianchi, F. D., De Battista, H., and Mantz, R. J. (2007). Wind turbine control systems - principles, modelling and gain scheduling design. London, United Kingdom: Springer-Verlag, London Limited.

Google Scholar

Bossanyi, E. A. (2000). The design of closed loop controllers for wind turbines. Wind Energy (Chichester). 3, 149–163. doi:10.1002/we.34

CrossRef Full Text | Google Scholar

Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM.

Google Scholar

Chilali, M., and Pascal, G. (1996). H design with Pole placement constraints: an LMI approach. IEEE Trans. Autom. Contr. 41, 358–367. doi:10.1109/9.486637

CrossRef Full Text | Google Scholar

Clemens, C., Gauterin, E., Pöschke, F., and Schulte, H. (2020). “Assessment criteria for the mechanical loads of wind turbines applied to the example of active power control, s. 1-7, berlin, Germany,” in Proceedings of IFAC World Congress 2020, Berlin, Germany, 341–347.

CrossRef Full Text | Google Scholar

Ekelund, T. (1994). “Speed control of wind turbines in the stall region,” in IEEE conference on control applications (Glasgow, United Kingdom), 227–232.

CrossRef Full Text | Google Scholar

Gauterin, E., Kammerer, P., Martin, K., and Schulte, H. (2015). Effective wind speed estimation: comparison between kalman filter and takagi-sugeno observer techniques. ISA-Transactions 62, 60–72.

PubMed Abstract | CrossRef Full Text | Google Scholar

Gauterin, E., Schulte, H., and Georg, S. (2014). Disturbance compensation by wind speed reconstruction based on a takagi-sugeno wind turbine model. J. Phys. Conf. Ser. 524, 012072. Proceedings ‘The Science of Making Torque from Wind 2014’. IOP Publishing. doi:10.1088/1742-6596/524/1/012072

CrossRef Full Text | Google Scholar

Georg, S. (2015). Fault diagnosis and fault-tolerant control of wind turbines nonlinear Takagi-Sugeno and sliding mode techniques. PhD thesis. Rostock, Germany: University Rostock, Fakultät für Maschinenbau und Schiffstechnik.

Google Scholar

Jena, D., and Rajendran, S. (2015). A review of estimation of effective wind speed based control of wind turbines. Renew. Sustain. Energy Rev. 43, 1046–1062. doi:10.1016/j.rser.2014.11.088

CrossRef Full Text | Google Scholar

Johansen, T. A., Shorten, R., and Murray-Smith, R. (2000). On the interpretation and identification of dynamic takagi-sugeno fuzzy models. IEEE Trans. Fuzzy Syst. 8 (3), 297–313. doi:10.1109/91.855918

CrossRef Full Text | Google Scholar

Jonkman, J. (2016). NWTC Design Codes (FAST). v8.16 edition. Golden, CO: National Renewable Energy Laboratory (NREL).

Google Scholar

Jonkman, J. B., and Jonkman, B. J. (2016). “Fast modularization framework for wind turbine simulation: full-system linearization,” in The science of making torque from wind, Vol. 753, 082010.

CrossRef Full Text | Google Scholar

Jonkman, J., Butterfield, S., Musial, W., and Scott, G. (2009). Definition of a 5-MW reference wind turbine for offshore system development. Technical report, NREL/TP-500-38060. Golden, Colorado: National Renewable Energy Laboratory.

Google Scholar

Jonkman, J. M., and Buhl, M. L. (2005). FAST user’s guide. Technical report, NREL/EL-500-38230. Golden, Colorado: National Renewable Energy Laboratory.

Google Scholar

Lendek, Z., Guerra, T. M., Babuška, R., and De Schutter, B. (2010). Stability analysis and nonlinear observer design using takagi-sugeno fuzzy models. Berlin: Springer-Verlag Berlin Heidelberg.

Google Scholar

Li, S., Yang, J., Chen, W.-H., and Chen, X. (2014). Disturbance observer-based control: Methods and applications. 1st edition. Boca Raton: CRC Press Taylor and Francis Group.

Google Scholar

Luenberger, D. G. (1971). An introduction to observers. IEEE Trans. Autom. Contr. 16, 596–602. doi:10.1109/tac.1971.1099826

CrossRef Full Text | Google Scholar

Luenberger, D. G. (1964). Observing the state of a linear system. IEEE Trans. Mil. Electron. 8, 74–80. doi:10.1109/tme.1964.4323124

CrossRef Full Text | Google Scholar

Lyapunov, A. M. (1992). The general problem of the stability of motion. Int. J. Control 55, 531–534. doi:10.1080/00207179208934253

CrossRef Full Text | Google Scholar

Ma, X.-J., Sun, Z.-Q., and He, Y.-Y. (1998). Analysis and design of fuzzy controller and fuzzy observer. IEEE Trans. Fuzzy Syst. 6 (1), 41–51. doi:10.1109/91.660807

CrossRef Full Text | Google Scholar

Ma, X., Poulsen, N. K., and Bindner, H. (1995). Estimation of wind speed in connection to a wind turbine. Technical report. The Technical University of Denmark.

Google Scholar

Ostergaard, K. Z., Brath, P., and Stoustrup, J. (2007). Estimation of effective wind speed. J. Phys. Conf. Ser. 75, 012082. doi:10.1088/1742-6596/75/1/012082

CrossRef Full Text | Google Scholar

Pöschke, F., Gauterin, E., Martin, K., Fortmann, J., and Schulte, H. (2020). Load mitigation and power tracking capability for wind turbines using a non-linear model-based controller. Wind Energy 23, 1792–1809.

Google Scholar

Pöschke, F., Gauterin, E., and Schulte, H. (2019). “Lmi region-based nonlinear disturbance observer with application to robust wind turbine control,” in New trends in observer-based control, emerging methodologies and applications in modelling, identification and control. Editors O. Boubaker, Q. Zhu, M. Mahmoud, J. Ragot, H. Reza Karimi, and J. Dávila 1 edition (Elsevier Books), 35–75.

Google Scholar

Pöschke, F., Petrović, V., Berger, F., Neuhaus, L., Hölling, M., Martin, K., et al. (2022). Model-based wind turbine control design with power tracking capability: a wind-tunnel validation. Control Eng. Pract. 120, 105014. doi:10.1016/j.conengprac.2021.105014

CrossRef Full Text | Google Scholar

Rauh, A., Dehnert, R., Romig, S., Lerch, S., and Tibken, B. (2021). Iterative solution of linear matrix inequalities for the combined control and observer design of systems with polytopic parameter uncertainty and stochastic noise. Algorithms 14, 205. doi:10.3390/a14070205

CrossRef Full Text | Google Scholar

Schlipf, D., Fischer, T., Carcangiu, C. E., Rossetti, M., and Bossanyi, E. (2010). “Load analysis of look-ahead collective pitch control using lidar,” in Deutsche Windenergiekonferenz DEWEK, Bremen, Germany.

Google Scholar

Schulte, H., and Gauterin, E. (2015). Fault-tolerant control of wind turbines with hydrostatic transmission using takagi-sugeno and sliding mode techniques. Annu. Rev. Control 40, 82–92. doi:10.1016/j.arcontrol.2015.08.003

CrossRef Full Text | Google Scholar

Takagi, T., and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man. Cybern. 15 (1), 116–132. doi:10.1109/tsmc.1985.6313399

CrossRef Full Text | Google Scholar

Tanaka, K., and Wang, H. O. (2001). Fuzzy control systems design and analysis: A linear matrix inequality approach. New York: John Wiley & Sons.

Google Scholar

Tanaka, K., Ikeda, T., and Wang, H. (1998). Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Trans. Fuzzy Syst. 6, 250–265. doi:10.1109/91.669023

CrossRef Full Text | Google Scholar

Tanaka, K., and Sano, M. (1994). A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer. IEEE Trans. Fuzzy Syst. 2 (2), 119–134. doi:10.1109/91.277961

CrossRef Full Text | Google Scholar

Tanaka, K., and Sano, M. (1994). “On the concept of fuzzy regulators and fuzzy observers,” in IEEE Conference on Fuzzy Systems, Orlando, FL, 767–772.

Google Scholar

Tanaka, K., and Sugeno, M. (1992). Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. 45 (2), 135–156. doi:10.1016/0165-0114(92)90113-i

CrossRef Full Text | Google Scholar

Tanaka, K., and Wang, H. O. (1997). “Fuzzy regulators and fuzzy observers: a linear matrix-inequality approach,” in IEEE International Conference on Decision and Control, San Diego, CA, 1315–1320.

Google Scholar

VanAntwerp, J. G., and Braatz, R. D. (2000). A tutorial on linear and bilinear matrix inequalities. J. Process Control 10, 363–385. doi:10.1016/s0959-1524(99)00056-6

CrossRef Full Text | Google Scholar

Wang, H. O., Tanaka, K., and Griffin, M. F. (1996). An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Syst. 4 (1), 14–23. doi:10.1109/91.481841

CrossRef Full Text | Google Scholar

Wang, H. O., Tanaka, K., and Griffin, M. F. (1995). “Parallel distributed compensation of nonlinear systems by takagi-sugeno fuzzy model,” in Proceedings of FUZZ IEEE/IFES’95, Yokohoma, Japan, 531–538.

Google Scholar

Yoneyama, J., Nishikawa, M., Katayama, H., and Ichikawa, A. (1998). Output stabilization of takagi sugeno fuzzy systems. FUZZY sets Syst. 111, 253–266. doi:10.1016/s0165-0114(98)00121-3

CrossRef Full Text | Google Scholar

Zemouche, A., Zerrougui, M., Boulkroune, B., Rajamani, R., and Zasadzinski, M. (2016). “A new lmi observer-based controller design method for discrete-time lpv systems with uncertain parameters,” in 2016 American Control Conference, Boston, MA, 2802–2807.

CrossRef Full Text | Google Scholar

Appendix

Specification of all steady operation points OPi

In Table A1 all steady Operation Points OPi of this work are listed.

Specification of the component submodels

In Tables A2A8, all matrices and steady states are listed, necessary to replicate the achieved results of this work with the help of NREL FAST 5MW reference wind turbine simulation software (see NREL (XXXX), Jonkman and Buhl (2005) and Jonkman et al. (2009)). Note: As the disturbing wind time series and excitation, respectively is restricted to wind speeds from v = 14m/s to v = 16m/s, the corresponding i submodels are restricted to i ∈ (Lendek et al., 2010; Jonkman and Jonkman, 2016) (see also explanation in Section 3.2).

TABLE A1
www.frontiersin.org

TABLE A1. States of the i steady state operations points OPi of the NREL FAST 5MW reference wind turbine with the wind speed vc,i, rotor rotational speed ωR,c,i, generator torque TG,c,i and pitch angle βc,i.

TABLE A2
www.frontiersin.org

TABLE A2. State matrices AiB and augmented state matrices ÃiB of the Blade model (for the submodels i ∈ [15,18]).

TABLE A3
www.frontiersin.org

TABLE A3. Input matrices BiB and augmented input matrices B̃iB of the Blade model (for the submodels i ∈ [15,18]).

TABLE A4
www.frontiersin.org

TABLE A4. Common output matrix CB and augmented common output matrix C̃B of the Blade model (for all submodels).

TABLE A5
www.frontiersin.org

TABLE A5. Steady states x̲c,iB and augmented steady states x̲̃c,iB of the Blade model (for the submodels i ∈ [15,18]).

TABLE A6
www.frontiersin.org

TABLE A6. Steady state pitch angle βc,i and generator torque TG,i (for the submodels i ∈ [15,18]).

TABLE A7
www.frontiersin.org

TABLE A7. State feedback matrices KiR of the (rigid body) Rotion drive train model (for the submodels i ∈ [15,18]).

TABLE A8
www.frontiersin.org

TABLE A8. Error state feedback gain matrices LiBw,j of the blade model based wind speed observers B for:- global Lyapunov approach with wA,E- local Lyapunov approach with wF,J- submodels i ∈ [15,18]- matrix elements j ∈ [1,3].

TABLE A9
www.frontiersin.org

TABLE A9. Analysis of the ultimate loads maxw and fatigue loads Seqw resulting from five different wind speed observers regarding the steady increase or decrease of the loads (evaluated separately for each of the two Lyapunov approaches with wA,E for the global wind speed observers and with wF,J for the local wind speed observers; based on the loads depicted in Figure 8).

TABLE A10
www.frontiersin.org

TABLE A10. Analysis of the ultimate loads maxw and fatigue loads Seqw resulting from five different wind speed observers regarding the steady increase or decrease of the loads (comparing both Lyapunov approaches with each other with wA,E for the global wind speed observers and with wF,J for the local wind speed observers; based on the loads depicted in Figure 8).

All states and parameters are expressed in SI units, despite the generator torque TG (that is expressd in kNm), pitch angles (that are expressed in (angular) degree) and the rotation speed (expressed in revolution per minute).

Specification of the LMI constraints

Restrictions for the decay rates αmin  / max and natural frequencies of the system response (characterised by the real part and imaginary part of the closed-loop poles Re(sP,i) and Im(sP,i)) can be specified with bounds, e.g. with horizontal lines (with αmax < Re(sP,i) < αmin)) and diagonal origin lines (with tan(θ)<Im(sP,i)Re(sP,i)θ<arctanIm(sP,i)Re(sP,i), also denoted as cone angle) in the complex pole map, resulting in bounded pole map segments and pole regions, respectively (Pöschke et al., 2019).

In (Chilali and Pascal, 1996), LMI representations for these bounded pole map segments, i.e. the restricted location of the poles and eigenvalues of a linear system, respectively, are derived.

For the upper vertical bound αB,max of the error dynamics, based on the blade design model, the following LMI holds (with Niw=PiwLiBw, defined in (20)):

AiBTPiCBTNiwT+PiAiBNiwCB2αB,maxPi,(23)

and for the lower vertical bound αB,min it holds:

AiBTPiCBTNiwT+PiAiBNiwCB2αB,minPi.(24)

For the cone angle bound θ of the error dynamics, based on the blade design model, the following LMI holds:

sinθAiBTPiCBTNiwT+PiAiBNiwCBcosθAiBTPi+CBTNiwT+PiAiBNiwCBcosθAiBTPiCBTNiwTPiAiB+NiwCBsinθAiBTPiCBTNiwT+PiAiBNiwCB0.(25)

The matrices used in (23) to (25) are listed in Tables A2, A4, A8.

All simulations were executed with the controller ct1210013. The following global Lyapunov approach based wind speed observers were used: Aot1210027, Bot1210030, Cot1210033, Dot1210051, Eot1210049, Fot1210036, Got1210039, Hot1210042, Iot1210043, Jot1210044.

The state observer ot1010001 was utilised.

For the wind excitation the Imp14.hh wind time series is used.

To calculate the mean Euclidian norm LiBw,j¯2 of the error-feedback gains LiBw,j [see (26)] and the average, mean Euclidian norm Li¯Bw,j¯2 [see (27)] the worksheet Uebersicht_L_Matrizen_Pitchwinkel-YYYY_MM_DD.xlsx is used.

Load analysis

For the ultimate loads maxw and fatigue loads Seqw, resulting from five different wind speed observers (i.e. for the wA,E globalwind speed observers and wA,E local wind speed observers; see Figure 8), the steady increase or decrease of the loads is evaluated separately for each of the two observer approaches (see Table A9) and in comparison to each other (see Table A10).

Keywords: global and local Lyapunov approach, Takagi–Sugeno framework, model-based controller and observer design, feedforward-feedback control, linear-matrix-inequality and pole region-based controller design, wind turbine application, elaborated wind turbine simulation model, load analysis

Citation: Gauterin E, Pöschke F and Schulte H (2023) Global Versus Local Lyapunov Approach Used in Disturbance Observer-Based Wind Turbine Control. Front. Control. Eng. 3:787530. doi: 10.3389/fcteg.2022.787530

Received: 30 September 2021; Accepted: 14 June 2022;
Published: 06 February 2023.

Edited by:

Andreas Rauh, University of Oldenburg, Germany

Reviewed by:

Robert Dehnert, University of Wuppertal, Germany
János Zierath, W2E Wind to Energy GmbH, Germany

Copyright © 2023 Gauterin, Pöschke and Schulte. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Eckhard Gauterin, gauterin@htw-berlin.de

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.