In this proceeding, a nonlinear system x ̲ ̇ = f ( x ̲ , u ̲ ) 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 (A _i , B _i ) (with i ∈ [1, i _max] and i _max ≡ Nr). These submodels comprise the state matrices A _i and input matrices B _i for the state vector x ̲ and input vector u ̲ and their steady state values x ̲ c , i and u ̲ c , i . In wind turbine application the nonlinearity mainly results from the aerodynamics of the rotor blades.

The linear submodels (A _i , B _i ) are blended in a convex sum x ̇ ̲ = ∑ i N r h i z ̲ A i x ̲ − x ̲ c , i + B i u ̲ − u ̲ c , i with y ̲ = C x ̲ and ∑ i N r h i z ̲ = 1 with 0 ≤ h i z ̲ ≤ 1 (1) with the help of membership functions h i ( z ̲ ) , which depend on the premise variable z ̲ . Within this proceeding, the membership functions h i ( z ̲ ) are defined by triangular-shaped weighting functions w _k,l (z _k ) (with the actual premise variable z _k (t) and z _k , respectively between the discrete linearisation points OP _i for z k l − 1 , z k l and z k l + 1 , see Figure 1) for each of the k ∈ [1, k _max] premise variables z _k (discretised in l (z _k ) ∈ [1, l _max (z _k )] linearisation points with w k , l z k = z k − z k l z k l − z k l − 1 if z k l − 1 < z k ≤ z k l 1 − z k − z k l z k l + 1 − z k l if z k l < z k ≤ z k l + 1 0 else. (2)

The weighting functions w _k,l (z _k ) are combined to form the ( i max ≡ ) N r = ∏ k = 1 k max l max ( z k ) membership functions h i ( z ̲ ) (with i ∈ [1, Nr]): ∑ i = 1 N r h i z ̲ t = ∏ k = 1 k max ∑ l = 1 l max z k w k , l z k . (3)

In this contribution, the membership functions h i ( z ̲ ) just blend the i submodels, which are direct adjacent to the actual operation point (with h i ( z ̲ ) ≠ 0 ), while the membership functions of all other models are set to zero (i.e., h i ( z ̲ ) = 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 v ̂ as the premise variable z ̲ ≡ v ̂ .

Within this contribution individual input-matrices B _i , e.g., depending on the wind speeds v and rotor rotation speed ω _R , and a common output-matrix C _i ≡C, just describing the measurable states y ̲ , are supposed.

In the TS framework, a state-space controller u ̲ = − K x ̲ with convexly blended state-feedback gains K _j —so-called Parallel Distributed Compensation (PDC) Controller (Wang et al. (1995), Tanaka and Wang (1997))—is usually used: u ̲ = − ∑ j N r h j z ̲ K j x ̲ − x ̲ c , j . (4) In Eq. 4 and the following, it is assumed, that z ̂ ̲ → z ̲ 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 u ̲ = − ∑ j N r h j z ̲ K j x ̲ − x ̲ c , j ⏟ u ̲ F B + ∑ j N r h j z ̲ u ̲ c , j ⏟ u ̲ F F = − ∑ j N r h j z ̲ K j x ̲ − x ̲ c , j − u ̲ c , j . (5) With the feedforward signal u ̲ F F the disturbance is attenuated, while the feedback signal u ̲ F B compensates control errors resulting from design model uncertainties.With Eq. 5 and ∑ i N r h i ( z ) = 1 in Eq. 1 the closed-loop dynamics (for i individual input matrices B _i , see Section 2.1.1) ¹ is described by x ̇ ̲ = ∑ i N r h i z ̲ A i x ̲ − x ̲ c , i + B i − ∑ j N r h j z ̲ K j x ̲ − x ̲ c , j − u ̲ c , j − u ̲ c , i = ∑ i N r ∑ j N r h i z ̲ h j z ̲ A i − B i K j x ̲ − x ̲ c , i . (6)

For nonlinear systems x ̲ ̇ = f ( x ̲ , u ̲ ) 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 L i ( y ̲ − y ̂ ̲ ) (Tanaka and Sano, 1994a). For a common output matrix C _i ≡C (see Section 2.1.1) it holds²: x ̲ ̂ ̇ = ∑ i N r h i z ̲ A i x ̲ ̂ − x ̲ c , i + B i u ̲ − u ̲ c , i + ∑ i N r h i z ̲ L i y ̲ − y ̲ ̂ y ̲ ̂ = C x ̲ ̂ = ∑ i N r h i z ̲ A i x ̲ ̂ − x ̲ c , i + B i u ̲ − u ̲ c , i + L i C x ̲ − x ̲ ̂ (7) with the reconstructed states x ̲ ̂ , the reconstructed outputs y ̲ ̂ and the error-feedback gains L _i .

For the error dynamics e ̇ ̲ e ̲ ≔ x ̲ − x ̂ ̲ ⇒ e ̇ ̲ = x ̇ ̲ − x ̂ ̇ ̲ (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 : e ̲ ̇ = ∑ i N r h i z ̲ A i x ̲ − x ̂ ̲ + L i C x ̲ − x ̲ ̂ = ∑ i N r h i z ̲ A i + L i C x ̲ − x ̲ ̂ ⏟ e ̲ . (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 v ̂ , that is used as the premise variable z ̲ , scheduling the feedforward and feedback signal (see Eq. 5 and Figure 3 with z ̲ = v ̂ ). 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 v ̂ as premise variable z ̲ , 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 z ̲ often comprises the reconstructed, disturbing wind speed v ̂ (within this contribution, the premise variable consists just of the reconstructed wind speed v ̂ , i.e. z ̲ = v ̂ holds), therefore the premise observer is also denoted as disturbance observer in the following (see also the observer classification in Section 2.4.3).

For the local Lyapunov approach the inequality derived for the closed-loop dynamics (with ( X ̲ , X ̲ ̇ ) ≡ ( x ̲ , x ̇ ̲ ) ) results in V ̇ = x ̲ T X i A i T − M i T B i T + A i X i − B i M i x ̲ < 0 with X i ≔ P i − 1 and M i = K i X i . (18) The inequality (Eq. 18) holds, if the LMI X i A i T − M i T B i T + A i X i − B i M i ≺ 0 (19) is satisfied, i.e., individual and local, respectively positive definite matrices X _i exist, which fulfill Eq. 19.

The same simplification holds for the error dynamics, i.e. for the local Lyapunov approach the inequality derived for the error dynamics (with ( X ̲ , X ̲ ̇ ) ≡ ( e ̲ , e ̇ ̲ ) ) results in: V ̇ = e ̲ T A i T P i − C T N i T + P i A i − N i C e ̲ < 0 with N i = P i L i . (20)

The inequality (Eq. 20) holds, if the LMI A i T P i − C T N i T + P i A i − N i C ≺ 0 (21) is satisfied, i.e. individual and local, respectively positive definite matrices P _i exist, which fulfill Eq. 21.

Once the individual matrix P _i and the slag parameter M _i (for local controller design) or N _i (for local observer design) is found with the LMI solver, the state-feedback gain K _i or error-feedback gain L _i is defined by K _i = M _i P _i or L i = P i − 1 N i . That is, the state-feedback gain K _i or error-feedback gain L _i 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 (A _i , B _i ) in an individual and single LMI, respectively (and not combining the dynamics of all or direct adjacent submodels).

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 P _i ≻ 0, see Eq. 17), the parallel and simultaneously, respectively to be solved LMIs are reduced (from n global,all∕adj. LMI,Cntrl + 1 > n global LMI,Obs ( = N r + 1 ) to n Local LMI,Cntrl∕Obs = 2 LMIs), too. Thereby, the LMI solver is executed Nr times consecutively for the local Lyapunov approach (due to the individual P _i -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.

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 P _i 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.

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 B lades, rotating as one rigid body in the rotor plane (denoted with rotor rotation (speed) ω R in Figure 2) and translating with x B 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 ).

After linearising an elaborated WT simulation model for i stationary operating points OP _i (with i ∈ [1, Nr]), the controller design submodels ( A i B / R , B i B / 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 ( A i B / R , B i B / R ) are given in Tables A2, A3 in the appendix.

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 T _G control, i.e., the generator torque T _G 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 T _R 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., u ̲ = [ β , T G ] 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 u ̲ c , i = [ β c , i , T G , c , i ] T for all 27 stationary operating points in partial and full load range are listed.

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 v _eff . 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 x B ³ 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 v ̂ ≡ v eff and blade translation speed x ̂ ̇ B from the blade translation error e x B determined from the measured and reconstructed blade translation e x B = x B − x ̂ B . For this purpose, the disturbance observer design model is augmented by a simple wind model (Ekelund, 1994), expanding the state vector x ̂ ̲ B with the reconstructed wind speed v ̂ (and disturbance signal d ̲ ≡ v ̂ , respectively) x ̃ ̂ ̲ B = [ x ̂ ̲ B , v ̂ ] T and augmenting the state space model Eq. 7 and Eq. 9 accordingly (with A ̃ i , B ̃ i and C ̃ ; 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 x ̂ ̲ B with the disturbing wind state v ̂ , is classified as an Extended State Observer (see (Li et al., 2014)), that is represented in the Takagi-Sugeno framework.

The reconstructed wind speed v ̂ is defined as premise variable z ̲ , scheduling the nonlinearity of the system and utilised for the membership function h i ( z ̲ ) of the feedforward signal u ̲ F F and feedback signal u ̲ F B 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-shaped ⁴ membership functions h _i (z) are used, just blending directly adjacent submodels (A _i , B _i ) (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 V _i is defined for each submodel (A _i , B _i ), because of the local stability condition Eq. 17. Therefore, the convex blending with the membership functions h i ( z ̲ ) 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 A ̃ i , B ̃ i and C ̃ , as well as the augmented steady system states x ̃ ̲ c , i and steady input states u ̲ c , i , used within this contribution, are given in Table A2 to Table A6. Additionally, the state feedback gain matrices K _i and error-feedback gain matrices L i B w are listed in Table A7 and Table A8.

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 (A _i , B _i ) into decoupled reduced submodels—denoted with component models—just comprising the dynamics of the individual, mechanical main components like the rotor blade submodels ( A i B , B i B ) and drive train submodels ( A i R , B i R ) (also described in (Pöschke et al., 2020), but now upgraded with the rotor blade submodels ( A i B , B i B ) ).

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 ( K i R , K i B ) and observer gains L i R , L i B ) are superposed for the overall controller gain K _i and observer gain L _i . The stability of the convexly blended i submodels is analysed with the superposed gains K _i and L _i 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 z ̲ (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 z ̂ ̲ , under the assumption of a maximum estimation error in the premise variable with z ̂ ̲ → z ̲ . Therefore, the stability proof and separation principle, respectively holds for z ̂ ̲ → z ̲ . Even though the maximum error estimation is missing in this contribution, it is assumed, that z ̂ ̲ → z ̲ 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 K _i and L _i , 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.

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 (A _i , B _i ) with i ∈ [15,18] are taken into account in the results and discussion.

Figure 4 shows the resulting p pole locations s P , i O L / 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   in A , E ) global wind speed observers and w ∈ F , J local 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 B lade model (with its two states x ̂ B , x ̂ ̇ B see Eq. 22) and the non-vibrating wind model (with the wind speed state v ̂ ), the design model posses three poles (i.e., p ∈ [1,3]): the conjugated-complex blade poles s P , i O L / w , 2 ∨ 3 (inside the pole map, for p = 2 ∨ p = 3) and the real-valued wind model pole s P , i O L / w , 1 (on the real axis of the pole map, for p = 1). In Table 2 the distances Δ s P , i w , p = Re s P , i O L , p − Re s P , i w , p 2 + Im s P , i O L , p − Im s P , i w , p 2 (23) between open-loop poles s P , i O L , p and closed-loop poles s P , i w , 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 Δ s P , i w , p ̄ = 1 3 ∑ p = 1 3 Δ s P , i w , p = 1 3 Δ s P , i w , 1 + Δ s P , i w , 2 + Δ s P , i w , 3 (24) of the single submodel i and the average distance Δ s P , i ̄ w , p ̄ = 1 4 ∑ i = 15 18 Δ s P , i w , p ̄ = 1 4 Δ s P , 15 w , p ̄ + Δ s P , 16 w , p ̄ + Δ s P , 17 w , p ̄ + Δ s P , 18 w , p ̄ (25) of all four submodels are listed in Table 2.

The related error-feedback gain matrices L i B w , j (with their j elements L i B w , j described in Eq. 22) are summarised in Table A8. To assess the L i B w , j deviations resulting from the shrinking pole regions, two different mean Euclidian norms ‖ L i ̄ B w , j ̄ ‖ 2 and ‖ L i ̄ B w , j ̄ 3 / j ̄ 1,2 ‖ 2 are calculated as metrics of the error-feedback gains L i , B w , j for each of the w ∈ A , E global wind speed observers and w ∈ F , J local wind speed observers and both metrics averaging all, four incorporated submodels i ∈ [15, 18]:

The metric ‖ L i ̄ B w , j ̄ ‖ 2 cumulates all three L i B w , j elements (with j ∈ [1, 3]) ‖ L i ̄ B w , j ̄ ‖ 2 = 1 4 ∑ i = 15 18 ‖ L i B w , j ̄ ‖ 2 = 1 4 ‖ L 15 B w , j ̄ ‖ 2 + ‖ L 16 B w , j ̄ ‖ 2 + ‖ L 17 B w , j ̄ ‖ 2 + ‖ L 18 B w , j ̄ ‖ 2 (26) with ‖ L i B w , j ̄ ‖ 2 = L i B w , 1 2 + L i B w , 2 2 + L i B w , 3 2 . The metrics ‖ L i ̄ B w , j ̄ 3 / j ̄ 1,2 ‖ 2 are calculated separately for L i B w , j ̄ 3 (to gain the wind speed state v ̂ ̇ , see Eq. 22) and L i B w , j ̄ 1,2 (to gain the blade states x ̂ ̇ B and x ̂ ̈ B ): ‖ L i ̄ B w , j ̄ 3 / j ̄ 1,2 ‖ 2 = 1 4 ∑ i = 15 18 ‖ L i B w , j ̄ 3 / j ̄ 1,2 ‖ 2 = 1 4 ‖ L 15 B w , j ̄ 3 / j ̄ 1,2 ‖ 2 + ‖ L 16 B w , j ̄ 3 / j ̄ 1,2 ‖ 2 + ‖ L 17 B w , j ̄ 3 / j ̄ 1,2 ‖ 2 + ‖ L 18 B w , j ̄ 3 / j ̄ 1,2 ‖ 2 with ‖ L i B w , j ̄ 3 ‖ 2 = L i B w , 3 2 and ‖ L i B w , j ̄ 1,2 ‖ 2 = L i B w , 1 2 + L i B w , 2 2 . (27) The results are depicted in Table 3.