Real-time implementation of nonlinear optimal-based vibration control for a wind turbine model

Abstract

In the present paper, a real-time implementation of a previously introduced nonlinear optimal-based vibration control method is presented. A vibrating structure, namely a wind turbine tower-nacelle laboratory model equipped with a tuned vibration absorber, is analysed. For control purposes, a magnetorheological damper is used in a tuned vibration absorber system. Force constraints of a magnetorheological damper are an intrinsic part of the implemented nonlinear technique. The aim of the current research is to experimentally investigate the influence of nonlinear optimal-based vibration control law quality index elements’ weights on the vibration attenuation effectiveness along with the magnetorheological damper stroke amplitude, maximum control current or force. As a reference, simple, optimal-based, modified ground-hook law with the sole control objective of primary structure deflection minimisation is used in addition to the passive systems with constant magnetorheological damper current values, proving the benefits of the proposed solution.

Keywords

Real-time vibration control nonlinear optimal-based control wind turbine magnetorheological damper tuned vibration absorber

Introduction

Vibrations may become a crucial problem for slender systems and structures such as towers, masts, chimneys, wind turbines,^1–7 bridges,^8,9 high buildings,^10,11 plate structures,^12,13 etc. Most of them are equipped with vibration attenuation or fatigue reduction solutions.

The concepts utilised to mitigate vibrations include tuned vibration absorbers (TVAs), tuned liquid (column) dampers, viscoelastic/hydraulic dampers, granular dampers, piezoelectric actuators, etc.^14,15 TVAs are widely used vibration reduction solutions. A standard (passive) TVA consists of an additional moving mass, a spring and a viscous damper, the parameters of which are tuned to the selected (most often first) mode of the vibration.¹⁶ Passive TVAs work well at the load conditions characterised with a single frequency but cannot adapt to a wide excitation spectrum.³ During the system/structure operation lifetime, its frequency spectra may vary; thus, more advanced TVA solutions are investigated to enable TVA tuning to the vibration frequency.

Another approach that may be used to cope with the vibration problem is the deployment of high strength and fatigue-reducing material technologies, throughout which nanotechnology seems one of the most promising. Copper nanoparticles can help to reduce the unevenness in the surface of the steel, which in turn reduces the amount of stress risers, limiting fatigue cracking. The addition of nanoparticles can help to solve the brittleness issue of the welds and improve the fracture problems associated with high strength bolts. Nano-aluminium-oxide of high purity improves the tensile and flexural strength of the concrete. Carbon nanoreinforments as well as nano-silica are also used to improve the concrete strength, with the latter increasing its resistance to water penetration too; nano-iron gives self-sensing capabilities along with improved compressive and flexible strengths. Zirconia nanocomposites exhibit superior physical (strength, hardness, and flexibility) and chemical endurance. The use of nano-coatings can result in higher corrosion and fire resistance, and also in self-cleaning capabilities.^17,18 These exemplary novel material solutions contribute to the increased lifespan of slender structures, due to the mechanism of nano-scale strength, hardness, and flexibility,^19,20 along with their smart-maintenance. On the other hand, there are several concerns related to nanotechnology, including high cost (vibration absorbers are less expensive and highly effective) as well as health and environmental hazards.

Instead of dealing with the sophisticated materials for vibration and fatigue minimisation, the current paper focuses on the advanced vibration absorbers. Among them, magnetorheological (MR) TVAs are placed,^5,6,11,21,22 as an MR damper guarantees a wide range of resistance force in comparison with a viscous damper and a high operational robustness along with minor energy requirements as compared with the active systems.^1,3,22–28 Simulations and experiments show that the implementation of an MR damper in the TVA system may lead to further vibration reduction with regard to passive TVA.^5,6

Most of the MR damper real-time control algorithms are based on the bang–bang approach or two-stage concepts with the calculation of an MR damper required force and precise force tracking algorithms.^9,22,29 The latter concepts suffer from the inability to produce the demanded MR damper force pattern due to, e.g. the impossibility to generate active forces and force value limitations: lower constraint imposed by the residual force and upper constraint imposed by the piston velocity and maximum MR damper current/magnetic circuit saturation. As a result, the force pattern calculated at the first stage is not the same at the output of the second stage. Moreover, some advanced first-stage algorithms need real-time frequency determination which may be an issue for polyperiodic or random vibrations.³⁰

Recently, an entirely different concept was proposed³⁰ – all of the MR damper force constraints are assumed to be an intrinsic part of a control method. This requires the utilisation of nonlinear control techniques, which include maximum principle-based methods,^4,13,31 control Lyapunov function-based methods⁸ including backstepping,^10,32 (feedback) linearization methods / linear optimal control theory based on the Riccati equation solution,^10,33 etc. Along with the advantages, each of the method groups have their disadvantages, that are significant for the current control problem consideration and solution.³⁰

The concept which real-time implementation is presented in the paper was originally developed for the vibration attenuation of systems/structures equipped with MR TVAs,³⁰ from among which, a wind turbine tower-nacelle system is investigated currently in particular, using its scaled model.^34–37 Wind turbines undergo external loads that vary in time, including wind load variations, wind shear, Karman vortices, the blade passing effect, differences in inflow conditions for each of the blades, sea waves and/or ice load, etc., as well as rotating elements’ unbalance, all of them contributing to the structural vibration and fatigue wear of towers and blades. The problem of wind turbine tower vibration was addressed within the scope of the Department’s research projects – a specially developed tower-nacelle laboratory model was built, assuming that a nacelle, a hub, a shaft, a generator, blades and possibly a gearbox are all represented by the rigid body fixed to the top of the vertical rod, representing a tower. The laboratory test rig gives the possibility to model tower vibrations under various aerodynamic, hydrodynamic, mechanical unbalance, changeable electromagnetic loads, etc., excitation sources – the horizontal concentrated force generated by a dedicated shaker may be applied at different heights of the rod, or at a nacelle position. Two initial bending modes of the vibration may be analysed; however, only the first one is investigated here and the MR TVA is tuned to its frequency. With the use of the MR damper, dedicated control solutions may be realized in comparison with the passive systems with constant MR damper current values.

The paper is organised as follows. First, a regarded system is depicted. Subsequently, the nonlinear optimal control problem is formulated and solved. Then, the implementation technique, the experimental setup and the test conditions are described. The paper is summarized with the real-time control results and several conclusions.

A regarded system

A wind turbine tower-nacelle model is considered as a primary vibrating system, which first bending mode (modal) mass is m₁, (modal) stiffness is k₁ and (modal) damping is c₁. A TVA of mass m₂, stiffness k₂ and damping c₂ is considered (Figure 1). The movement of both m₁ and m₂ is restricted to be linear displacement x₁ and x₂ (respectively) along the common axis (horizontal in Figure 1) of an applied excitation force $P$ . A mass m₂ is assumed to be 7.69% of a mass m₁. Stiffness k₂ and damping c₂ parameters are tuned according to the standard Den Hartog approach.¹⁶ For control purposes, passive damping c₂ is replaced by an MR damper of RD1097–1 series.²⁵ The values of the adopted system parameters are presented in Table 1.

Figure 1.

Two-body diagram of a regarded system with an MR TVA.

Table 1.

The adopted system parameters.

Parameter	Value
m ₁	168.29 kg
k ₁	82 554 N/m
c ₁	49.53 Ns/m
m ₂	12.94 kg (7.69% m₁)
k ₂	5717 N/m
c ₂	82.64 Ns/m

Problem formulation and solution

Consider the equation of a vibrating system with an MR TVA

\dot{z} (t) = f (z (t), u (t), t), t \in [t_{0}, t_{1}]

(1)

where

z (t)

is a state vector

z (t) = {[\begin{matrix} \begin{matrix} z_{1} (t) & z_{2} (t) \end{matrix} & \begin{matrix} z_{3} (t) & z_{4} (t) \end{matrix} \end{matrix}]}^{T}

(2)

u (t)

is a piecewise-continuous, constrained scalar control function (

u (t) \in U

) and a quality index to be minimised is

G (z, u) = \int_{t_{0}}^{t_{1}} g (z (t), u (t), t) d t

(3)

Following ‘A regarded system’ section, assume: $z_{1} = x_{1}$ , $z_{2} = {\dot{x}}_{1}$ , $z_{3} = x_{2}$ , $z_{4} = {\dot{x}}_{2}$ , thus

f (z (t), u (t), t) = [\begin{matrix} \begin{matrix} z_{2} (t) \\ \frac{1}{m_{1}} (- (k_{1} + k_{2}) z_{1} (t) - c_{1} z_{2} (t) + k_{2} z_{3} (t) + F_{m r} (z (t), u (t), t) + P (t)) \end{matrix} \\ \begin{matrix} z_{4} (t) \\ \frac{1}{m_{2}} (k_{2} z_{1} (t) - k_{2} z_{3} (t) - F_{m r} (z (t), u (t), t)) \end{matrix} \end{matrix}]

(4)

where

\begin{matrix} F_{m r} (z (t), u (t), t) = (C_{1} i_{m r} (u (t), t) + C_{2}) \tanh {ν [(z_{4} (t) - z_{2} (t)) + (z_{3} (t) - z_{1} (t))]} \\ + (C_{3} i_{m r} (u (t), t) + C_{4}) [(z_{4} (t) - z_{2} (t)) + (z_{3} (t) - z_{1} (t))] \end{matrix}

(5)

is the MR damper force according to the hyperbolic tangent model with parameters:

C_{1}

C_{2}

C_{3}

C_{4}

ν

,³⁸ while

i_{m r} (u (t), t)

is the MR damper coil current, and

P (t)

is the excitation force applied to the primary structure. To account for the control constraints, i.e. the MR damper current limitation to

[0, i_{m a x}]

range (

i_{m a x} > 0

), it was further assumed³⁰

i_{m r} (u (t), t) = i_{m a x} \sin^{2} (u (t))

(6)

The considered quality function is

g (z (t), u (t), t) = g_{11} z_{1}^{2} (t) + g_{12} z_{2}^{2} (t) + g_{13} {(z_{1} (t) - z_{3} (t))}^{2} + g_{21} i_{m r}^{2} (u (t), t) + g_{22} F_{m r}^{2} (z (t), u (t), t)

(7)

to account for the primary structure displacement

z_{1}

and velocity

z_{2}

minimisation, the MR damper stroke

z_{1} - z_{3}

minimisation, and the MR damper control current

i_{m r}

and force

F_{m r}

minimisation.

Assume the Hamiltonian in the form

H (ξ (t), z (t), u (t), t) = - g (z (t), u (t), t) + ξ^{T} (t) f (z (t), u (t), t)

(8)

If $(z^{*} (t), u^{*} (t))$ is an optimal controlled process, there exists an adjoint (co-state) vector function $ξ$ satisfying the equation

\dot{ξ} (t) = - f_{z}^{* T} (z^{*} (t), u^{*} (t), t) ξ (t) + g_{z}^{T} (z^{*} (t), u^{*} (t), t), t \in [t_{0}, t_{1}]

(9)

with a terminal (transversality) condition

ξ (t_{1}) = 0

(10)

so that

u^{*} (t)

maximises the Hamiltonian over the set

U

for almost all

t \in [t_{0}, t_{1}]

(

f_{z}

and

g_{z}

are

f

and

g

derivatives with respect to

z

;

f

and

g

are assumed to be continuously differentiable with respect to the state and continuous with respect to time and control).³⁹ For the regarded system, the co-state vector is

ξ (t) = {[\begin{matrix} \begin{matrix} ξ_{1} (t) & ξ_{2} (t) \end{matrix} & \begin{matrix} ξ_{3} (t) & ξ_{4} (t) \end{matrix} \end{matrix}]}^{T}

(11)

while

\begin{matrix} f_{z}^{* T} (z^{*} (t), u^{*} (t), t) = [\begin{matrix} 0 & - \frac{1}{m_{1}} (k_{1} + k_{2} + {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t)) & 0 & \frac{1}{m_{2}} (k_{2} + {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t)) \\ 1 & - \frac{1}{m_{1}} (c_{1} + {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t)) & 0 & \frac{1}{m_{2}} {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t) \\ 0 & \frac{1}{m_{1}} (k_{2} + {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t)) & 0 & - \frac{1}{m_{2}} (k_{2} + {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t)) \\ 0 & \frac{1}{m_{1}} {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t) & 1 & - \frac{1}{m_{2}} {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t) \end{matrix}] \end{matrix}

(12)

with

\begin{matrix} {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t) = ν (C_{1} i_{m r} (u^{*} (t), t) + C_{2}) {1 - \tanh^{2} [ν (z_{4}^{*} (t) + z_{3}^{*} (t) - z_{2}^{*} (t) - z_{1}^{*} (t))]} \\ + (C_{3} i_{m r} (u^{*} (t), t) + C_{4}) \end{matrix}

(13)

thus

\begin{array}{l} {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t) = \frac{\partial F_{m r} (z^{*} (t), u^{*} (t), t)}{\partial z_{3}^{*} (t)} = \frac{\partial F_{m r} (z^{*} (t), u^{*} (t), t)}{\partial z_{4}^{*} (t)} \\ = - \frac{\partial F_{m r} (z^{*} (t), u^{*} (t), t)}{\partial z_{1}^{*} (t)} = - \frac{\partial F_{m r} (z^{*} (t), u^{*} (t), t)}{\partial z_{2}^{*} (t)} \end{array}

(14)

and

\begin{array}{l} g_{z}^{T} (z^{*} (t), u^{*} (t), t) \\ = {[\begin{matrix} \begin{matrix} _{g}^{z 1} (z^{*} (t), u^{*} (t), t) & 2 g_{12} z_{2}^{*} (t) - 2 g_{22} F_{m r}^{′} (z^{*} (t), u^{*} (t), t) \end{matrix} & \begin{matrix} - 2 g_{13} (z_{1}^{*} (t) - z_{3}^{*} (t)) + 2 g_{22} F_{m r}^{′} (z^{*} (t), u^{*} (t), t) & 2 g_{22} F_{m r}^{′} (z^{*} (t), u^{*} (t), t) \end{matrix} \end{matrix}]}^{T} \end{array}

(15)

where

\begin{array}{l} _{g}^{z 1} (z^{*} (t), u^{*} (t), t) = 2 g_{11} z_{1}^{*} (t) + 2 g_{13} (z_{1}^{*} (t) - z_{3}^{*} (t)) - 2 g_{22} F_{m r}^{′} (z^{*} (t), u^{*} (t), t) \\ F_{m r}^{′} (z^{*} (t), u^{*} (t), t) = F_{m r} (z^{*} (t), u^{*} (t), t) {\tilde{F}}_{m r} (z^{*} (t), u^{*} (t), t) \end{array}

The Hamiltonian maximisation condition³⁹ is

\begin{array}{l} \frac{\partial H (ξ (t), z^{*} (t), u (t), t)}{\partial u (t)} \\ = {(\frac{1}{m_{1}} λ_{2} (t) - \frac{1}{m_{2}} λ_{4} (t) - 2 g_{22} F_{m r} (z^{*} (t), u (t), t)) \frac{\partial F_{m r} (z^{*} (t), u (t), t)}{\partial i_{m r} (u (t), t)} - 2 i_{m a x} g_{21} \sin^{2} (u (t))} \\ \times \sin (2 u (t)) i_{m a x} = 0 \end{array}

(16)

with the appropriate sign change regime, where

\frac{\partial F_{m r} (z^{*} (t), u (t), t)}{\partial i_{m r} (u (t), t)} = C_{1} \tanh [ν (z_{4}^{*} (t) + z_{3}^{*} (t) - z_{2}^{*} (t) - z_{1}^{*} (t))] + C_{3} (z_{4}^{*} (t) + z_{3}^{*} (t) - z_{2}^{*} (t) - z_{1}^{*} (t))

For the assumed $u (t)$ range: $U = [0, π)$ , equation (16) results in

\sin (2 u (t)) = 0

\sin^{2} (u (t)) = \frac{1}{2 i_{m a x} g_{21}} (\frac{1}{m_{1}} ξ_{2} (t) - \frac{1}{m_{2}} ξ_{4} (t) - 2 g_{22} F_{m r} (z^{*} (t), u (t), t)) \frac{\partial F_{m r} (z^{*} (t), u (t), t)}{\partial i_{m r} (u (t), t)}

(17)

Analogically to Martynowicz³⁰

\begin{array}{l} i_{m r}^{*} (u^{*} (t), t) \\ = {\begin{array}{l} 0, & i f R H S (17) < 0 \\ \frac{1}{2 g_{21}} (\frac{1}{m_{1}} ξ_{2} (t) - \frac{1}{m_{2}} ξ_{4} (t) - 2 g_{22} F_{m r} (z^{*} (t), u (t), t)) \frac{\partial F_{m r} (z^{*} (t), u (t), t)}{\partial i_{m r} (u (t), t)}, & i f R H S (17) \in [0 1) \\ i_{m a x}, & i f R H S (17) \geq 1 \end{array} \end{array}

(18)

where

R H S (17)

is the right-hand side of equation (17).

The implementation technique

The common approach to optimal control of nonlinear systems is offline computation of the optimal control $u^{*} (t)$ for the problem (1)(3)(8)÷(10). However, so calculated open loop control suffers from lack of robustness to uncertainties (e.g. unmodelled dynamics, perturbations of external forces or initial conditions), and thus perturbation control techniques are used.^13,31 However, proper linearisation may be an issue for highly nonlinear systems with implicit relations between state, co-state and control.

Regarding the above considerations concerning uncertain systems, the boundary value problem (1)(3)(8)÷(10) may be solved for every sample period of a real-time implementation. Due to the high computational load required online, the optimisation horizon may be assumed to be equal to one integration step; however, this also requires performance data acquisition, computation and control (DAC) environment when dealing with complex nonlinear systems. Utilising the MATLAB/Simulink environment, the bvp4c iterative scheme based on the three-stage Lobatto IIIa collocation formula⁴⁰ may be used for an efficient solution of the implicit nonlinear boundary value problems. Alternative iterative approaches are the Adomian decomposition method (ADM), the variational iteration method (VIM),⁴¹ the homotopy analysis method (HAM)⁴² and the homotopy method coupling with the perturbation technique – the homotopy perturbation method (HPM),⁴³ etc. HAM provides a family of solution expressions in the auxiliary parameter (even if a nonlinear problem has a unique solution). The auxiliary parameter gives an additional way to adjust the convergence region and rate of solution series; however, there are no rigorous theories to choose the auxiliary conditions.⁴⁴ In contrary, HPM uses the auxiliary parameter as a small parameter, and only few iterations are needed, unlike HAM where an infinite series is necessary.⁴⁵ Similar to HPM, VIM produces rapidly convergent successive approximations without the restrictive assumptions. It has been widely employed due to its flexibility, convenience and accuracy. Regarding the original VIM method,⁴¹ many additional merits have been discovered and some advancements are suggested.^46,47 Thus, VIM and HPM methods can effectively solve a large class of nonlinear problems including those with strong nonlinearity; they do not impose the limitations or assumptions required in classical perturbation methods, while they can overcome the difficulties arising in handling the ADM and HAM methods.^45,48,49

Recently, a numerical implementation of one-step nonlinear optimal control for systems with MR TVAs was realised with MATLAB/Simulink dedicated level-2 s-function and bvp4c algorithm;³⁰ it was also shown that iteration procedures implicating large computational loads may be omitted, utilising very short time range basis optimal problem tasks with a high-frequency resetting function and zero initial conditions for all co-state integrators. In such a case, it was demonstrated that the influence of the transversality condition (10) error was negligible for the regarded MR TVA(s) control application.³⁰ Thus, a real-time implementation of nonlinear optimal-based vibration control, with all of its advantages (including the various possible elements of the quality index), using simple DAC environment is possible.

For the purpose of the real-time control of the wind turbine tower-nacelle model first bending mode, the approach described in the ‘Problem formulation and solution’ section and in ref.³⁰ was implemented. The system and the initial MR damper model parameters as in Table 1 and Maślanka et al.³⁸ were used along with the MR damper control current formula (18) (this solution is further designated by Opt4). After the completion of the identification of the particular MR damper unit used, hyperbolic tangent model (5) parameters were modified accordingly (see Table 2) and utilised for the verified algorithm version, designated further by Opt7, being the basis for the analysis of various control cases I–IV (see ‘The test conditions’ section).

Table 2.

The identified MR damper model parameters.

Parameter	Value
C ₁	44
C ₂	1.0
C ₃	225
C ₄	7.0
$ν$	70

A modified two-level displacement ground-hook law (Mod.GND) – the simple implementation of the optimal control for the case when primary system/structure displacement amplitude minimisation is the sole objective^5,6,30 – was additionally tested during the present experimental study. This control law switches the control current between 0 and $i_{m a x}$ with regard to $x_{1}$ and $F_{m r}$ signs. A basic version of this algorithm with direct $x_{1}$ and $F_{m r}$ measurements was designated by Mod.GND1, while the version with $F_{m r}$ correction (accounting for the MR-damper–force-sensor hold inertia) was designated by Mod.GND2. Using the experimental basis, the Mod.GND1&2 approaches were compared with the Opt4 and Opt7 techniques. For the latter, quality index (7) weights were set either to minimise the primary system displacement, or the primary system displacement along with the MR damper stroke (to account for a damper stroke restriction), or the primary system displacement along with the MR damper control current (to account for a damper current limitation due to, e.g. thermal constraints)/the MR damper force (to account for a damper force limit).

The experimental setup

The analysed wind turbine tower-nacelle laboratory model had to fulfil various constraints imposed by the laboratory facility and project limitations. It was assumed that at least a partial dynamic similarity (similarity of motions of tower tips) between a real-world wind turbine tower-nacelle system (Vensys 82) and its scaled model has to be fulfilled.^34,35,37 The absorber mass $m_{2}$ was selected to be 7.69% of the modal mass $m_{1}$ of the tower-nacelle model first bending mode (see Table 1). The stiffness of the TVA was also attuned to the first bending mode according to Den Hartog.¹⁶

The test rig (Figure 2) consists of a vertically oriented titanium alloy circular rod (no. 1, representing the wind turbine tower) and a system of steel plates (no. 2, representing the nacelle-assembly including the turbine) fixed to the top of the rod, with the MR TVA embedded. The titanium rod is rigidly mounted to a steel foundation frame (no. 3). The MR TVA is an additional mass (no. 4) moving horizontally along linear bearing guides, connected with the assembly representing the nacelle via springs (no. 5) and RD-1097–1 MR damper (no. 6) in parallel. The MR TVA operates along the same direction as the vibration excitation applied. The structure is excited by the TMS2060E lightweight electrodynamic shaker (no. 7 (TMS⁵⁰)), the force of which is applied to the system of steel plates (no. 2) modelling the nacelle with the help of the drive train assembly (no. 8) of the changeable leverage.

The DAC system consists of a laser transducer for the nacelle-assembly displacement (i.e. the tower deflection) $x_{1}$ measurement, a laser vibrometer for the MR TVA mass absolute displacement $x_{2}$ measurement, the MR damper/TMS2060E force sensors, the MR damper current transducer, as well as a supply-conditioning system for the inputs/outputs, and a measuring-control PC equipped with Inteco multi I/O board of the RT-DAC4 series (InTeCo Ltd.⁵¹) and MATLAB/Simulink/RT-CON applications along with Real-Time Workshop library and Real Time Windows Target extension. With the help of the RT-CON software, real-time regimes are satisfied using the 1 kHz sampling frequency basis. The RT-CON analogue input drivers are used to enter the measurement signals into the Simulink model. The MATLAB/Simulink control algorithm calculates the demanded MR damper current in each sample step and outputs it with the use of the RT-CON/Simulink analogue output driver and the RT-DAC4 analogue output channel. This signal is then amplified with the use of the dedicated circuit equipped with the PID analogue control loop to force the required electric current value through the MR damper coil. Another measurement-control PI loop is implemented using the MATLAB/Simulink/RT-CON/RT-DAC4 environment for the generation of the demanded excitation force to be compared with the TMS2060E force transducer signal, and conditioned by the TMS 2100E21-400 dedicated amplifier.^34–37

The test conditions

The test conditions parameters are as follows. The wind turbine tower-nacelle model is excited by a harmonic force of amplitude $A (P (t)) = 45.4$ N and frequency range of [2.5, 4.5] Hz. The fixed sample step $t_{s} = 10^{- 3}$ s is adopted. The weighting factors for the quality index (7) are assumed to (control cases I–IV):

minimise the primary structure deflection (nacelle-assembly displacement) $x_{1}$ amplitude as the sole objective (being also the primary objective for the cases II, III and IV): $g_{11} = 10^{18}$ , $g_{12} = 0$ , $g_{13} = 0$ , $g_{21} = 4$ , $g_{22} = 0$ ;

minimise additionally the MR damper stroke amplitude: $g_{11} = 10^{18}$ , $g_{12} = 0$ , $g_{13} = 10^{16}$ vs. $g_{13} = 10^{18}$ , $g_{21} = 4$ , $g_{22} = 0$ ;

minimise additionally the MR damper current: $g_{11} = 10^{18}$ , $g_{12} = 0$ , $g_{13} = 0$ , $g_{21} = 4 \cdot 10^{9}$ , $g_{22} = 0$ ( $i_{m a x} = 1.0$ A was assumed, as the MR damper current minimisation objective assured safe operation according to Lord Rheonetic 25 );

minimise additionally the MR damper force: $g_{11} = 10^{18}$ , $g_{12} = 0$ , $g_{13} = 0$ , $g_{21} = 4$ , $g_{22} = 10^{7}$ (using the modelled $F_{m r}$ value in equation (18), based on equation (5) and $x_{1}$ , $x_{2}$ and $i_{mr}$ measurements) vs. $g_{22} = 10^{11}$ (using the directly measured $F_{m r}$ value in equation (18)). The value of $i_{m a x} = 1.0$ A was assumed (the MR damper force minimisation objective assured safe operation according to Lord Rheonetic²⁵).

While the MR damper current or force minimisation was not regarded as one of the primary quality objectives (i.e. control cases I and II), continuous current limitation $i_{m a x} = 0.5$ A was assumed with respect to Lord Rheonetic.²⁵ The same limit was adopted for the Mod.GND law (versions 1 and 2).

Real-time control results

The efficiency of the adopted solutions is analysed using the frequency characteristics of the dynamic amplification factor (DAF)

D A F = \frac{A (x_{1} (t))}{A (P (t)) / k_{1}}

(19)

the MR damper stroke amplitude

A (x_{1} - x_{2}

) and the MR damper force amplitude MRF (with the frequency resolution of 0.1 Hz within [3.0, 4.0] Hz range and 0.2 Hz outside this range), along with the time patterns of

x_{1}

x_{1} - x_{2}

i_{m r}

, and

F_{m r}

. Figures 3, 6 and 9 present the frequency characteristics of, respectively, DAF,

A (x_{1} - x_{2})

, and MRF, obtained for the system with the Opt4 and Opt7 implementations of the optimal-based control and the Mod.GND law (versions 1 and 2), with regard to the passive system with constant MR damper current values of 0.000 A, 0.125 A and 0.250 A. Similarly, Figures 4, 7 and 10 present the frequency patterns of, respectively, DAF,

A (x_{1} - x_{2})

, and MRF, obtained for the Opt7 solution with different

g_{13}

(an MR damper stroke weight) values. Additionally, Figures 5, 8 and 11 present the frequency characteristics of, respectively, DAF,

A (x_{1} - x_{2})

, and MRF, obtained for the Opt7 concept with different

g_{21}

(an MR damper current weight) and

g_{22}

(an MR damper force weight) values, with regard to the minimum current/force solution (0.00 A).

Figure 2.

The laboratory test rig: (a) a general view, (b) the MR TVA.

Figure 3.

Dynamic amplification factor DAF frequency characteristics: passive system vs. Mod.GND1&2 vs. Opt4&7.

Figure 4.

Dynamic amplification factor DAF frequency characteristics: Opt7 for different $g_{13}$ values.

Figure 5.

Dynamic amplification factor DAF frequency characteristics: Opt7 for different $g_{21}$ and $g_{22}$ values.

Figure 6.

The MR damper stroke amplitude $A (x_{1} - x_{2}$ ) frequency characteristics: passive system vs. Mod.GND1&2 vs. Opt4&7.

Figure 7.

The MR damper stroke amplitude $A (x_{1} - x_{2}$ ) frequency characteristics: Opt7 for different $g_{13}$ values.

Figure 8.

The MR damper stroke amplitude $A (x_{1} - x_{2}$ ) frequency characteristics: Opt7 for different $g_{21}$ and $g_{22}$ values.

Figure 9.

The MR damper force amplitude MRF frequency characteristics: passive system vs. Mod.GND1&2 vs. Opt4&7.

Figure 10.

The MR damper force amplitude MRF frequency characteristics: Opt7 for different $g_{13}$ values vs. 0.00 A.

Figure 11.

The MR damper force amplitude MRF frequency characteristics: Opt7 for different $g_{21}$ and $g_{22}$ values vs. 0.00 A.

Figures 12 to 14 present the comparison of the primary structure deflection $x_{1}$ , the MR damper piston displacement $x_{1} - x_{2}$ , the MR damper control current $i_{m r}$ and force $F_{m r}$ time patterns obtained for the specific frequency points: 3.2 Hz, 3.5 Hz and 3.6 Hz of the frequency characteristics shown in Figures 3 to 11. To improve the clarity of comparison, all the respective axes of the time responses have the same limits.

As may be observed (Figure 3), an MR damper force correction algorithm (Mod.GND2) improves the attenuation properties by more precise $F_{m r}$ sign detection. Similarly, identification of the particular MR damper unit model parameters, that are used in the adjoint dynamic equation (9), (12) and (15) and in the control current formula (18), is a root of the Opt7 advantage over the Opt4 method. Both Mod.GND2 and Opt7 provide maximum DAF values close to 2.0, which is encouraging, assuming the 7.69% mass ratio TVA real-time implementation (that is significantly better than DAF for any passive solution regarded). This is obtained at ca. 6.5 mm MR damper stroke amplitude (Figure 6) and ca. 11 N maximum MR damper force (Opt7), whereas Mod.GND2 requires more than 16 N maximum force (Figure 9).

Observing Figures 4, 7, 10, and 12, one may note that the MR damper stroke amplitude $A (x_{1} - x_{2}$ ) for $g_{13} = 10^{16}$ is slightly reduced (within [3.4, 3.9] Hz range) or similar to $g_{13} = 0$ case with almost no disadvantages concerning the DAF characteristics; however, extra MR damper force MRF is required, especially in [3.4, 3.9] Hz range (Figures 4, 7, 10, and 12(a) and (b)). For $g_{13} = 10^{18}$ case, significant DAF and MRF increase is observed (Figures 4, 7, 10, and 12(a) and (c)); however, $A (x_{1} - x_{2}$ ) is greatly reduced with respect to $g_{13} = 0$ and $g_{13} = 10^{16}$ cases; thus, the control objective of $A (x_{1} - x_{2}$ ) minimisation is attained for most frequencies.

Similarly, observing Figures 5, 8, 11, 12(a) and (d), 13(a) and (b), and 14(a) and (b), it is evident that by increasing $g_{21}$ weight from the value of $4$ to $4 \cdot 10^{9}$ , one obtains the reduction of control current value and/or duty cycle with no need of $i_{m a x}$ limitation to $0.5$ A (what is essential for the control cases I and II, ‘The test conditions’ section, with regard to Lord Rheonetic²⁵ – see also Remark below). Moreover, the control patterns obtained for Opt7, $g_{21} = 4 \cdot 10^{9}$ are characterised with needle-shape impulses that are set at proper time instants, which is proven to be the most effective MR damper control method. In parallel, at 3.5 Hz (which is the first bending frequency of the regarded system without TVA), no control action is produced ( $i_{m r} = 0$ ) for Opt7, $g_{21} = 4 \cdot 10^{9}$ , which is considered as the best control approach at this frequency and should result in the coincidence of the 0.00 A and Opt7, $g_{21} = 4 \cdot 10^{9}$ points at DAF and MRF frequency characteristics – that is not the case due to the residual force resulting from the magnetic remanence present during the Opt7, $g_{21} = 4 \cdot 10^{9}$ experiment, while not during the 0.00 A test (this may be eliminated by negative current spikes as in Martynowicz²²); note that the residual force is visibly smaller in Figure 14(b) than in (a).

Remark

For the control cases I and II, no truncate effect at $i_{m a x} = 0.5$ A may be observed, as the measured $i_{m r}$ patterns are presented – sharp truncation is clear for the demanded MR damper current profiles.

Using the $g_{21}$ weight, it is possible to lower the MRF (and DAF) frequency characteristics at the primary structure resonance frequency; however, at some other frequency ranges, MRF is not as low as for $g_{22} = 10^{7}$ (using the modelled $F_{m r}$ value) or $g_{22} = 10^{11}$ (using the measured $F_{m r}$ value), see Figures 11 and 13(c) and (d) vs. 13(b). The $g_{22}$ weight is designed to reduce an MR damper force in a wide range, while still delivering the satisfactory DAF and $A (x_{1} - x_{2}$ ) ratings.

Figure 12.

Time responses at 3.6 Hz: (a) Opt7, (b) Opt7, $g_{13} = 10^{16}$ , (c) Opt7, $g_{13} = 10^{18}$ , (d) Opt7, $g_{21} = 4 \cdot 10^{9}$ , (e) Opt7, $g_{22} = 10^{7}$ ( $F_{m r}$ modelled), (f) Opt7, $g_{22} = 10^{11}$ ( $F_{m r}$ measured); all unmentioned weights as for control case I (‘The test conditions’ section).

Figure 13.

Time responses at 3.2 Hz: (a) Opt7, (b) Opt7, $g_{21} = 4 \cdot 10^{9}$ , (c) Opt7, $g_{22} = 10^{7}$ ( $F_{m r}$ modelled), (d) Opt7, $g_{22} = 10^{11}$ ( $F_{m r}$ measured); all unmentioned weights as for control case I (‘The test conditions’ section).

Figure 14.

Time responses at 3.5 Hz: (a) Opt7, (b) Opt7, $g_{21} = 4 \cdot 10^{9}$ ; all unmentioned weights as for control case I (‘The test conditions’ section).

Recapitulating, by increasing the $g_{13}$ value, reduced stroke amplitude (relevant for the systems with restricted working space/MR damper stroke) may be obtained; by increasing the $g_{21}$ weight, effective control is obtained with the limited value and/or duty cycle of the current, while additionally the control signal is lowered to zero if necessary – because of this, good performance is attainable for an MR damper type characterised with maximum current restriction (as for e.g. RD-1097–1); finally, by increasing the $g_{22}$ weight, satisfactory vibration attenuation is available when using an MR damper with smaller maximum force (ca. 8 N in the current task).

Conclusions

The aim of the current research was a real-time implementation and experimental test of the nonlinear optimal-based vibration control technique for a system/structure (namely a wind turbine tower-nacelle model) equipped with an MR TVA. Using a simple hardware, the previously developed technique³⁰ was successfully implemented and verified, including additional optimisation fields covered by a nonlinear optimal control quality index, in comparison with simple yet reliable (optimal-based) modified ground-hook law, and passive solutions with constant MR damper current. The obtained maximum DAF values of ca. 2.0 are more than adequate for the assumed 7.69% TVA mass ratio, proving the effectiveness and validity of the control approach.³⁰ Neither MR damper force tracking nor oscillation frequency determination is necessary online; thus, control quality is not compromised by MR damper force constraints, while real-time control during transient, polyperiodic or random vibration phases is unimpeded; also neither offline calculation nor disturbances assumption is necessary, all essential for continual real-time control.

The experimental tests proved that the Opt7 method delivers the most valuable vibration control results, especially in comparison with a reference modified two-level displacement ground-hook law that previously demonstrated one of the best vibration reduction properties.^5,6,22,30 However, the Mod.GND algorithm is devoted for the case when only the primary system displacement amplitude has to be minimised, while the proposed solution copes well with different constraints (optimisation fields) as, e.g. working space, MR damper stroke, current or force, that may be encountered during the operation of real-world systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by AGH University of Science and Technology (research program no. 11.11.130.766); and the Polish National Science Centre (NCN) (decision no. DEC-2013/11/D/ST8/03387).

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