Suppression of the vibrations of wind turbine towers

Transcription

1 IMA Journal of Mathematical Control and Information , doi:1.193/imamci/dnr14 Advance Access publication on June 27, 211 Suppression of the vibrations of wind turbine towers XIAOWEI ZHAO Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Corresponding author: AND GEORGE WEISS Department of Electrical Engineering-Systems, Tel Aviv University, Ramat Aviv 69978, Israel [Received on 15 February 211; revised on 15 April 211; accepted on 23 May 211 We investigate the suppression of the vibrations of a wind turbine tower using colocated feedback to achieve strong stability. We decompose the system into one model describing the vibrations in the plane of the turbine axis and another model in the plane of the turbine blades. In the plane of the turbine axis, we obtain a non-uniform Spacecraft Control Laboratory Experiment SCOLE model with either force control or torque control. We show that this system is strongly stabilizable by static output feedback from either the velocity or the angular velocity of the nacelle. In the plane of the turbine blades, we model the wind turbine tower as a non-uniform SCOLE system coupled with a two-mass drive-train model, the control being the torque created by the electrical generator. We prove that generically, this system can be strongly stabilized by feedback from the angular velocity of the nacelle and the angular velocity of the generator rotor. Keywords: wind turbine tower model; SCOLE model; strong stabilization. 1. Introduction The aim of this paper is to investigate the suppression of the vibrations of wind turbine towers. As a source of renewable and clean energy, wind power is rapidly increasing its share of the grid capacity in many countries. Large offshore turbines are subjected to severe weather causing vibrations leading to fatigue. Their life expectancy is difficult to evaluate, but it is important to keep it high. This implies a need for a control system to suppress the vibrations of the turbine tower without affecting the reliability of the power supply. To design a vibration suppression controller, we need to find a suitable model for the wind turbine tower and then to study its controllability and stabilization. The current wind turbine tower models are mostly based on finite-element approximations, which divide the tower into many slices and consider these slices as interconnected finite-dimensional systems. These models are very complex and are used mainly for simulation. We decompose the system into one model describing the vibrations in the plane of the turbine axis and another model in the plane of the turbine blades. These models are distributed parameter models of the tower. c The author 211. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 378 X. ZHAO AND G. WEISS Think of a wind turbine tower clamped in the ocean floor that carries at its top the nacelle with a weight of several 1 tons. This heavy nacelle has a big effect on the wind turbine tower; we cannot ignore it and simply model the tower as a beam. The NASA Spacecraft Control Laboratory Experiment SCOLE system is a well-known model for a flexible beam with one end clamped and the other end linked to a rigid body. It has been originally developed to understand the dynamics of a mast carrying an antenna on a satellite see Littman & Markus, 1988a,b. In the vertical plane of the turbine axis, it is sensible to model the wind turbine tower as a non-uniform SCOLE system with either force control or torque control. The force control can be obtained by modulating the turbine pitch angle, while the torque control can be obtained by an electrically driven mass located in the nacelle. First, we discuss stabilization in the vertical plane of the turbine axis. When the wind acts on the blades, there are two kinds of effects in this plane. One effect is the force that acts on blades. We use this for force control by modulating the turbine pitch angle this is called pitch angle control. The other effect is the torque acting on the nacelle and the blades. Compared to the force, this effect is small and we consider it as a disturbance. For our purpose of stabilizing the tower in the plane of the turbine axis, we can ignore the blades and the hub as a separate rotating body and just include them as part of the nacelle. As the diameter of the tower decreases with height and it is partially submerged in water, it can be described by a non-uniform SCOLE model studied in Guo 22, denoted by Σ d : ρxw tt x, t + EIxw xx x, t xx =, < x < l, t >, w, t =, w x, t =, mw tt l, t EIw xx x l, t = f t, Jw xtt l, t + EIlw xx l, t = vt, where the subscripts t and x denote derivatives with respect to the time t and the position x, respectively. w stands for the transverse displacement of the tower, l is the height of the tower and EI and ρ are the flexural rigidity function and mass density function. m > and J > are the mass and the moment of inertia of the nacelle. EIxw xx x, t xx dx is the total lateral force acting on a slice of the tower of height dx, located at the position x and the time t. EIw xx x l, t and EIlw xx l, t are the force and the torque acting on the nacelle from the tower at the time t. f and v are the force control and the torque control acting on the nacelle. In a practical engineering design, we would probably only use force control. But for the sake of generality, we cover also the case of torque control. We assume that ρ, EI C 4 [, l are strictly positive functions. The natural state at the time t of this wind turbine tower model Σ d is 1.1 zt = [z 1, t z 2, t z 3 t z 4 t, 1.2 where the superscript means transpose, and z 1 x, t = wx, t, z 2 x, t = w t x, t, z 3 t = w t l, t, z 4 t = w xt l, t. The energy state space of Σ d is Here, X = H 2 l, l L2 [, l C H 2 l, l ={h H2, l h =, h x = }, 1.4

3 SUPPRESSION OF THE VIBRATIONS OF WIND TURBINE TOWERS 379 where H n n N denote the usual Sobolev spaces. The norm on X squared is zt 2 = EIx z 1xx x, t 2 dx + ρx z 2 x, t 2 dx + m z 3 t 2 + J z 4 t 2, 1.5 which represents twice the physical energy. Exponential stability is the most desirable kind of stability. In Rao 1995, using an energy multipliers method, it is proved that the exponential stabilization of the uniform SCOLE model can be obtained by high-order output feedback. In Guo 22, it was shown that the exponential stabilization of the non-uniform SCOLE model can also be achieved by high-order output feedback using the time derivative of the strain at the end of the beam. However, with high-order output feedback, the closed-loop system is not well-posed. In addition, such a feedback is difficult to realize in practice. That is why velocity and angular velocity are more natural signals for the stabilization of the SCOLE model. Unfortunately, it is proved in Rao 1995 using a method of compact perturbation that the uniform SCOLE model is not exponentially stabilizable by boundary feedback from the velocity and the angular velocity of the rigid body. Therefore, also the non-uniform SCOLE model cannot be exponentially stabilizable ieneral by using these signals. Strong stabilization becomes a good compromise to aim for. The strong stabilization of the uniform SCOLE model in one specific case was proven in Littman & Markus 1988a. We are not aware of any strong stabilization result for the non-uniform SCOLE model. In the first main result of this paper, we show that the wind turbine tower model Σ d i.e., the nonuniform SCOLE model is strongly stabilizable on X using either torque control or force control with colocated feedback from either the velocity or the angular velocity of the nacelle see Theorem 3.3. In the plane of the turbine blades, we have to use a more complex model. When the wind acts on the blades in the plane of the turbine blades, the force acting on the nacelle is negligible and can be considered as a disturbance. However, the gearbox transfers most of the torque caused by the wind to the nacelle. Our model must take the gearbox and the flexible low-speed turbine shaft into account. We use a two-mass drive-train model see Fig. 1 studied in Hansen et al. 22 to analyse the mechanical system within the nacelle. For a wind turbine tower moving in the plane of the turbine blades, the model Σ c is ρxw tt x, t + EIxw xx x, t xx =, x, t, l [,, 1.6 w, t =, w x, t =, 1.7 mw tt l, t EIw xx x l, t = f t, 1.8 Jw xtt l, t + EIlw xx l, t = T lss t + T hss t b m θ m t t T e t, 1.9 θ T tt t = 1 Ta t T lss t w xtt l, t, 1.1 θ m tt t = 1 J G Thss t b m θ m t t T e t + w xtt l, t, 1.11 θ k t = θ T t θ mt, 1.12 T lss t = K s θ k t + C s θ k t t = T hss t, 1.13

4 38 X. ZHAO AND G. WEISS FIG. 1. The two-mass drive-train model with gearbox. T a is the active torque from the turbine and T e is the electric torque of the generator, which acts between the rotor connected to the high speed shaft and the stator connected to the nacelle. where the equations are a non-uniform SCOLE model Σ d as in 1.1. T hss and T lss are the torque acting on the gearbox from the high speed shaft and the low speed shaft, respectively. f is the force control, which can be obtained by an electrically driven mass located in the nacelle. T e is the electric torque control created by the electrical generator. The meaning of other notation and assumptions are the same as in 1.1. In this paper, we consider only torque control for Σ c. Equations are a two-mass drive-train model. θ T and θ m are the angles of the turbine rotor and generator rotor with respect to the nacelle. θ k is the angular difference between the endpoints of the low speed shaft. > is the rotational inertia of the turbine blades and other low speed components e.g. the hub, while J G > is the rotational inertia of the generator rotor. K s > and C s are the torsional stiffness and torsional damping coefficient of the low speed shaft. b m is the damping coefficient of the high speed shaft. T a is the active torque from the turbine, which is a disturbance from our point of view. is the gearbox ratio. To derive this model, we have taken into account w xtt l,, the angular acceleration of the nacelle: θ T tt t+w xtt l, t and θ m tt t w xtt l, t are the angular accelerations of the turbine rotor and generator rotor with respect to the earth. We assume that > J G, which is always true in reality e.g. from Wang, 29, Table 3.1, we can see that is much bigger than J G. We allow b m and C s, so that our results do not rely on physical damping. The natural state at the time t of this wind turbine tower model Σ c is z c t = [zt qt, 1.14 where zt is the state of the subsystem Σ d at the time t, as in 1.2, while qt = [q 1 t q 2 t q 3 t is the state of the two-mass drive-train model at the time t. Here, q 1 t = θ T t t + w xt l, t, q 2 t = θ m t t w xt l, t and q 3 t = θ k t. The energy state space of Σ c is X c = X C 3 = H 2 l, l L2 [, l C 5. The natural norm on X c squared is z c t 2 = zt 2 + q 1 t 2 + J G q 2 t 2 + K s q 3 t 2, where zt 2 from 1.5 is the squared natural norm on X, while the last three terms represent qt 2. The number z c t 2 represents twice the physical energy of Σ c. The second main result of this paper is the following generic strong stabilization result: for every choice of the strictly positive functions ρ n, EI n C 4 [, l and of the parameters l >, m n >, J n >,

5 SUPPRESSION OF THE VIBRATIONS OF WIND TURBINE TOWERS 381 K s >, >, J G >, >, C s and b m, there are at most three values μ> such that the system Σ c with ρ = μρ n, EI = μei n, m = μm n, J = μj n is not strongly stabilizable on X c, by the static output feedback T e = ky c + v, where v is the new input typically zero and y c t = 1 J w xtl, t 1 J G θ m t t w xt l, t. The feedback gain k can be any positive number see Theorem 4.3. We make some remarks about alternatives to the Euler Bernoulli beam model used in 1.1 and again in 1.6. This beam model is derived from Fx, t = M x x, t, ρxw tt x, t = F x x, t, Mx, t = EIxw xx x, t, where F is the horizontal force acting on a cross section of the beam at height x by the beam above, and M is the torque acting on the same cross section from the beam above. There are many alternative beam models, depending on what physical phenomena we want to model and to what accuracy. For instance, the first equation above which, when multiplied with dx, expresses the equilibrium of torques acting on a small segment of the beam may be replaced with the more accurate Fx, t = M x x, t Pxw x x, t + αxw ttx x, t, 1.15 where Px is the weight of the beam plus nacelle above the height x and αxdx is the moment of inertia of a segment of height dx of the beam. The equation 1.15, multiplied with dx, is just Newton s law for rotational movement applied to a segment of the beam. Including only the term containing α would lead to a Rayleigh beam model. Whichever model we choose, one may question why we have not used a more accurate one, and whether our results say anything about the true system. We cannot answer such questions, as they are beyond mathematics. An analysis similar to the one here may be tried for more accurate models. This paper is divided into four sections. Section 2 is dedicated to the background on a class of infinite-dimensional linear systems with semigroup generator, bounded control operator and bounded observation operator. We recall the concepts of exact controllability, approximate controllability, stability and stabilization. We also introduce a proposition concerning the stabilization of such systems having a contraction semigroup and colocated control and observation. In Section 3, we derive the state space formulation of the wind turbine tower model in the plane of the turbine axis Σ d, described by 1.1. We derive a feedback law via either velocity through force control or angular velocity through torque control of the nacelle that achieves strong stabilization. In Section 4, we derive the state space formulation of the wind turbine tower model in the plane of the turbine blades Σ c, described by We obtain its generic strong stabilization via a feedback from the angular velocity of the nacelle and the angular velocity of the generator rotor through torque control.

6 382 X. ZHAO AND G. WEISS 2. Feedback stabilization for a class of infinite-dimensional linear systems Let the infinite-dimensional linear system Σ with input space U, state space X and output space Y be described by the equations {żt = Azt + But, 2.1 yt = Czt. Here, A is the generator of a strongly continuous semigroup T on X, B is a bounded control operator i.e. B LU, X and C is a bounded observation operator i.e. C LX, Y. The functions u L 2 loc [, ; U, z C[,, X and y C[,, Y are the input signal, state trajectory and output signal, respectively. The transfer function of Σ is Gs = CsI A 1 B. The basic reference for such systems is Curtain & Zwart We define the operators Φ τ for τ> by where u L 2 loc [, ; U. Φ τ u = τ T τ t Butdt, DEFINITION 2.1 The system Σ or the pair A, B is said to be exactly controllable if Ran Φ τ = X for some finite time τ >. The system Σ or the pair A, B is said to be approximately controllable if Ran Φ τ is dense in X for some finite time τ>. DEFINITION 2.2 The system Σ or the semigroup T is called weakly stable if T t z, y ast, for all z, y X. The system Σ or the semigroup T is called strongly stable if T t z ast for all z X. The system Σ or the semigroup T is called exponentially stable if its growth bound is negative. Static output feedback stabilization of the system Σ means finding a feedback operator K LY, U to make the system stable in some sense with the input u = Ky + v, where v is the new input function. The closed-loop system is described by 2.1, but with A + BKC in place of A. This operator A + BKC generates the closed-loop semigroup T K.IfK exists such that T K is exponentially, or strongly, or weakly stable, then we call the original system exponentially, or strongly, or weakly stabilizable by static output feedback, respectively. From Benchimol 1978, Theorem 3.1 and Corollary 3.1, we have the following proposition: PROPOSITION 2.3 Let Σ be as in 2.1 with A being maximal dissipative, Y = U and C = B.IfΣ is approximately controllable, then Σ is weakly stabilizable by the static output feedback u = ky + v, for any number k >, where v is the new input. Furthermore, if A has compact resolvents, then Σ is strongly stabilizable by the same static output feedback. Note that in Benchimol 1978, Theorem 3.1 and Corollary 3.1, k = 1, but it is easy to see that any k > will work. Actually, strictly positive operators would work as well. 3. Strong stabilization of a wind turbine tower model in the plane of the turbine axis In this section, we analyse the strong stabilization of the wind turbine tower model in the plane of the turbine axis, Σ d described by 1.1. As mentioned in Section 1, the state of Σ d at the time t is zt = [z 1 t z 2 t z 3 t z 4 t = [w, t w t, t w t l, t w xt l, t. 3.1

7 SUPPRESSION OF THE VIBRATIONS OF WIND TURBINE TOWERS 383 Here, z 1 and z 2 are the transverse displacement and the transverse velocity of the beam, respectively. z 3 and z 4 are the velocity and angular velocity of the nacelle, respectively. The natural energy state space of Σ d is X from 1.3. The natural norm on X has been defined in 1.5. We define the generator A as follows: ξ 2 ρ Aξ = 1 xeixξ 1xx x xx m 1 EIξ 1xx x l, ξ DA, 3.2 DA = J 1 EIlξ 1xx l { } ξ [H 4, l Hl 2, l H2 l, l ξ 3 = ξ 2 l C2. ξ 4 = ξ 2x l Let B 1 = [ m 1 which corresponds to only force control and B2 = [ 1 J which corresponds to only torque control. We use either force control or torque control so that our control operator B is either B 1 or B 2. We use a colocated sensor, which means that we take B as our observation operator. Then the wind turbine tower model in the plane of the turbine axis 1.1 can be rewritten as {żt = Azt + But, yt = B 3.3 zt. It is clear that B and B are bounded, i.e. B LC, X, B LX, C. In the case of force control, the output signal is 1 m times the velocity of the nacelle. In the case of torque control, the output signal is 1 J times the angular velocity of the nacelle. PROPOSITION 3.1 The generator A of the wind turbine tower model in the plane of turbine axis Σ d described by 3.3 is skew-adjoint on the state space X and ρa. Proof. According to Tucsnak & Weiss 29, Proposition 3.7.3, to show Proposition 3.1, it is enough to show that A is skew-symmetric and onto. First, we show that A is skew-symmetric, i.e. Re Aξ,ξ = for all ξ = [ξ 1 ξ 2 ξ 3 ξ 4 DA: ξ 2 ξ 1 ρ Re Aξ,ξ =Re 1 xeixξ 1xx x xx m 1 EIξ 1xx x l, ξ 2 ξ 2 l J 1 EIlξ 1xx l ξ 2x l = Re = Re + Re + Re EIxξ 2xx xξ 1xx x EIxξ 1xx x xx ξ 2 x dx EIξ 1xx x lξ 2 l EIlξ 1xx lξ 2x l EIxξ 1xx xξ 2xx x EIxξ 1xx x xx ξ 2 x dx EIξ 1xx x lξ 2 l EIlξ 1xx lξ 2x l. 3.4

8 384 X. ZHAO AND G. WEISS Using integration by parts, and the boundary conditions ξ 2 = and ξ 2x =, we get EIxξ 1xx xξ 2xx x EIxξ 1xx x xx ξ 2 x dx = [ EIxξ 1xx xξ 2x x l EIxξ 1xx x x ξ 2x xdx [ EIxξ 1xx x x ξ 2 x l + EIxξ 1xx x x ξ 2x xdx = EIlξ 1xx lξ 2x l EIξ 1xx x lξ 2 l. 3.5 Substituting 3.5 into 3.4, we get Re Aξ,ξ =, so that A is skew-symmetric. We still have to show that A is onto, i.e. Ran A = X, which means that for any fixed n = [n 1 n 2 n 3 n 4 X, the equation Aξ = n with ξ DA has a solution. The equation Aξ = n is equivalent to ξ 2 x = n 1 x, 3.6 ρ 1 xeixξ 1xx x xx = n 2 x, 3.7 m 1 EIξ 1xx x l = n 3, 3.8 J 1 EIlξ 1xx l = n From 3.6, it is easy to see that ξ 2 = n 1,ξ 3 = n 1 l, ξ 4 = n 1x l. What left is to determine ξ 1. Denote x by α α, l in 3.7 and multiplying by ρα from its both sides, we get EIαξ 1αα α αα = ραn 2 α. Integrating both sides of the above equation from β for any β, ltol, we get Combining this result with 3.8, we have EIξ 1αα α l EIβξ 1ββ β β = ραn 2 αdα. β EIβξ 1ββ β β = β ραn 2 αdα + mn 3. Integrating both sides of the above equation from γ for any γ, ltol,weget EIlξ 1ββ l EIγ ξ 1γγ γ = γ β ραn 2 αdα dβ + mn 3 l γ. Combining this result with 3.9, we get ξ 1γγ γ = EI 1 γ ραn 2 αdα dβ EI 1 γ mn 3 l γ EI 1 γ Jn 4. γ β

9 SUPPRESSION OF THE VIBRATIONS OF WIND TURBINE TOWERS 385 Integrating both sides of the above equation from to δ for any δ, l, using the boundary condition ξ 1γ =, integrating both sides of the resulted equation from to x and then using the boundary condition ξ 1 =, finally, we get x δ ξ 1 x = EI 1 γ ραn 2 αdα dβ dγ dδ γ β x δ x δ EI 1 γ mn 3 l γdγ dδ EI 1 γ Jn 4 dγ dδ. PROPOSITION 3.2 The resolvents β I A 1 β ρa are compact. Proof. From well-known Sobolev embedding theorems such as Proposition in Tucsnak & Weiss, 29 and similar results, we know that I LDA, X is a compact operator if we regard DA as a Hilbert space with the graph norm. Since β I A 1 LX, DA for any β ρa, it follows that I β I A 1 : X X is compact. THEOREM 3.3 The wind turbine tower model in the plane of the turbine axis Σ d described by 3.3 is strongly stabilizable by static colocated output feedback with any gain k >, from either the velocity or the angular velocity of the nacelle. Proof. From Proposition 3.1, we know that the generator A of Σ d is skew-adjoint on the state space X. It is clear that the control and observation operators B and B are bounded on X. Thus, Σ d described by 3.3 fits the framework of Proposition 2.3. From Guo & Ivanov 25, Theorems 3.2 and 2.3, it follows that Σ d is approximately controllable with only force control or only torque control on X. From Proposition 3.2, we know that the generator A has compact resolvents. Therefore, by Proposition 2.3, it follows that Σ d is strongly stabilizable by static output feedback with any gain k > from either the velocity or the angular velocity of the nacelle. 4. Strong stabilization of the wind turbine tower model in the plane of the turbine blades In this section, we analyse the strong stabilization of the wind turbine tower model in the plane of the turbine blades Σ c, described by We only use torque control here. As we mentioned in Section 1, the natural state of Σ c at the time t is z c t = [zt qt, where z is the state of Σ d as introduced at the beginning of Section 3, while qt = [q 1 t q 2 t q 3 t is the state of the two-mass drive-train model at the time t. Recall that q 1 t = θ T t t + w xt l, t and q 2 t = θ m t t w xt l, t are the angular velocities of the turbine rotor and generator rotor with respect to the earth and q 3 t is the angular difference between the endpoints of the low speed shaft. The energy state space of Σ c is X c = X C 3 = H 2 l, l L2 [, l C The norm on X c squared is [ ξ ς 2 = ξ 2 + ς 2

10 386 X. ZHAO AND G. WEISS for all ξ = [ξ 1 ξ 2 ξ 3 ξ 4 X and for all ς = [ς 1 ς 2 ς 3 C 3, where ξ 2 is as in 1.5 and ς 2 = ς J G ς K s ς 3 2. The state space formulation of Σ c is {żc t = A c z c t + B c u e t, y c t = B c z c t, 4.2 where u e = T e is the control input. For every [ ξ ς DA c,wehave A c [ ξ ς = EIl ξ 2 ρ 1 xeixξ 1xx x xx m 1 EIξ 1xx x l J ξ 1xx l + K s1+ J ς 3 b m J ς 2 + ξ 4 + C s1+ J κ, K s ς 3 C s κ K s J G ς 3 b m JG ς 2 + ξ 4 + C s J G κ κ where κ = ς 1 n 1 g ς 2 1+ ξ 4. From Zhao & Weiss 211, we know that DA c = DA C 3 = {[ ξ ς } [H 4, l Hl 2, l H2 l, l ξ 3 = ξ 2 l C ξ 4 = ξ 2x l Note that we use colocated sensors and actuators, meaning that the observation operator of Σ c has been chosen to be the adjoint of the control operator. We have B c = [ 1 J 1 [, B c = 1 J G J To prove our main result in this section, first we need to show a proposition: 1 J G. PROPOSITION 4.1 Let A b : DA b X be maximal dissipative. Let Q LX and A = A b + Q with DA = DA b. We have the following results: 1 If A is dissipative on X, then A is maximal dissipative on X. 2 If A b has compact resolvents, then A has compact resolvents. 3 If A b has compact resolvents and A is skew-symmetric on X, then A is skew-adjoint on X. Proof. A is a bounded perturbation of A b, hence it is a semigroup generator. Therefore, for sufficiently large β >, β I A has a bounded inverse hence it is onto. Hence, if A is dissipative, then A is maximal dissipative on X see, e.g. Tucsnak & Weiss, 29, Definition Thus, statement 1 holds.

11 SUPPRESSION OF THE VIBRATIONS OF WIND TURBINE TOWERS 387 Now, we show statements 2 and 3. Since A b has compact resolvents, and A has a non-empty resolvent set since it is a semigroup generator, also A has compact resolvents. Hence, σa consists of isolated points, so that si A is invertible for almost every s <. The facts that A is skew-symmetric and both β I A and β I + A are onto for some β>implies that A is skew-adjoint on X see, e.g. Tucsnak & Weiss, 29, Proposition PROPOSITION 4.2 The generator A c of the wind turbine tower model in the plane of the turbine blades Σ c described by 4.2 is maximal dissipative with compact resolvents on the state space X c. If the damping coefficients C s and b m are zero, then A c is skew-adjoint. Proof. First, we show that A c is dissipative, which means that Re A c [ ξ ς, [ ξς for all [ ξς DA c, i.e., for all ξ DA and ς C 3.Wehave Re A c [ ξ ς = Re, [ ξ ς EIl ξ 2 ρ 1 xeixξ 1xx x xx m 1 EIξ 1xx x l J ξ 1xx l + K s1+ J ς 3 b m J ς 2 + ξ 4 + C s1+ J κ, K s ς 3 C s κ K s J G ς 3 b m JG ς 2 + ξ 4 + C s J G κ ς 1 n 1 g ς 2 1+ ξ 4 By Proposition 3.1, we have Re Aξ,ξ =. Using this in the above equation, we get Re [ [ A c ξ ξ Ks 1 +, = Re ς ς + Re + Re ς 3 ξ 4 b m ς 2 ξ 4 b m ξ 4 K s ς 3 ς 1 C s κς 1 + K s K s ς 1 ς 3 K s ς 2 = b m ξ 4 + ς 2 2 C s ς 1 1 ς ξ 4 n g ξ 4 + C s1 + ξ 1 ξ 2 ξ 3 ξ 4 ς 1 ς 2 ς 3. κξ 4 ς 3 ς 2 b m ς 2 ς 2 b m ξ 4 ς 2 + C s κς 2 ς 3 K s1 + ξ 4 ς It can be verified see Theorem 3.2 in Zhao & Weiss, 211 that [ A c A Bd = f C Bc = b f C a [ A + [ Bd f C Bc, b f C a

12 388 X. ZHAO AND G. WEISS where A is as in 3.2, B = [ 1, J C = [ 1, a = C s C s J G C s C s + b J G n 2 m g JG K s K s J G 1 1, b f = C s1+ C s 1+ + b J G n 2 m g JG 1+, [ Cs 1+ c = C s1+ K b s 1+ n 2 m g, d f = C s1 + 2 n 2 + b m. g [ A It is easy to see that is skew-adjoint with compact resolvents on X c as A has these properties [ Bd on X, according to Propositions 3.1 and 3.2 and that f C Bc is bounded on X b f C a c. From 4.4, we know that A c is dissipative. Thus, by Proposition 4.1, it follows that A c is maximal dissipative with compact resolvents. From 4.4, we know that A c is skew-symmetric in the case that b m = and C s =. Again by Proposition 4.1, we conclude that in this case A c is skew-adjoint. THEOREM 4.3 For every choice of the strictly positive functions ρ n, EI n C 4 [, l and of the parameters l >, m n >, J n >, K s >, >, J G >, >, C s and b m, there are at most three values μ> such that the wind turbine tower model in the plane of the turbine blades Σ c described by 4.2 with ρ = μρ n, EI = μei n, m = μm n, J = μj n is not strongly stabilizable on the state space X c from 4.1 by a static output feedback u e = ky c + v where y c t = B c z c t = 1 J w xtl, t 1 J G θ m t t w xt l, t and v is the new input function. The feedback gain k can be any positive number. Proof. From Proposition 4.2, we know that the generator A c of Σ c is maximal dissipative with compact resolvents on the state space X c. Clearly, the control and observation operators B c and B c are bounded from C to X c and from X c to C. Thus, Σ c described by 4.2 fits the framework of Proposition 2.3. From Zhao & Weiss 211, we know that Σ c is exactly controllable on the state space DA c for all except three values of μ. This implies the approximate controllability of Σ c on X c for all but three values of μ. According to Proposition 2.3, it follows that Σ c is strongly stabilizable subject to the conditions in Theorem 4.3 by the static output feedback u e = ky c + v, for any feedback gain k >. Funding The first author s research was supported by an Engineering and Physical Sciences Research Council International Doctoral Scholarship an Overseas Research Students Awards Scheme Award and a British Council Researcher Exchange Programme Award, during his Ph.D. study at Imperial College, London. The research of the second author in the area of coupled systems is supported by the grant 71/1 of the Israel Science Foundation.

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