Classical Flutter and Active Control of Wind Turbine Blade Based on Piezoelectric Actuation

Abstract

The aim of this article is to analyze classical flutter and active control of single-cell thin-walled composite wind turbine blade beam based on piezoelectric actuation. Effects of piezoelectric actuation for classical flutter suppression on wind turbine blade beam subjected to combined transverse shear deformation, warping restraint effect, and secondary warping are investigated. The extended Hamilton's principle is used to set up the equations of motion, and the Galerkin method is applied to reduce the aeroelastic coupled equations into a state-space form. Active control is developed to enhance the vibrational behavior and dynamic response to classical aerodynamic excitation and stabilize structures that might be damaged in the absence of control. Active optimal control scheme based on linear quadratic Gaussian (LQG) controller is implemented. The research provides a way for rare study of classical flutter suppression and active control of wind turbine blade based on piezoelectric actuation.

Academic Editor:Mario Terzo

College of Mechanical & Electronic Engineering, Shandong University of Science & Technology, Qingdao 266590, China

Received 18 April 2015; Revised 7 July 2015; Accepted 7 July 2015; 29 July 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Although instability of stall nonlinear flutter has generally been an important issue in flutter research [1], classical flutter was observed involving a high-speed rotating blade in pitch excitation process. Simultaneously, with the advent of large wind turbine fitted with relatively slender blades, classical flutter may become a more important design consideration. In addition, innovative blade designs involving the use of aeroelastic tailoring and structural tailoring, wherein the blade twists as it bends under the action of aerodynamic loads to shed load resulting from wind turbulence, may increase the blade's proclivity for flutter [2].

Most of the literature of classical flutter focused on the helicopter blades, noncoupling wing sections, and fan rotor blades [3-5]. As for active control, most of the literature focused on beam structures rather than the blade sectional shapes. Kapuria and Yasin [6] study the active vibration suppression of hybrid composite and fiber metal laminate plates integrated with piezoelectric fiber reinforced composite sensors and actuators. Phung-Van et al. [7] present an effective formulation based on higher-order shear deformation theory to investigate free vibration and dynamic control of piezoelectric composite plates integrated with sensors and actuators. Supersonic flutter control of a three-layered sandwich curved panel of rectangular plan form with an adaptive electrorheological fluid core layer is investigated by Hasheminejad and Motaaleghi [8].

Park and Kim [9] study the active twist rotor blade incorporating single crystal macrofiber composite actuators and analyze the aeroelasticity. Although the rotor blade properties dynamically represent a real rotor blade, the analytical objects, are the torsional behavior of helicopter blades or the vibration behavior of airfoils in which only the torsional motions are involved. Simultaneously, the objects of the existing literature mostly concentrate on the frequency research rather than time response analysis.

In this work the classical flutter and flutter suppression of composite blade beam are investigated for single-cell thin-walled structure with piezoelectric patch embedded. The validity of the piezoelectric actuation is tested and illustrated by time domain response analysis rather than frequency research. The analysis is applied to a laminated host structure of the circumferentially asymmetric stiffness (CAS) that produces bending-twist-transverse shear coupling. The governing system can be derived by the extended Hamilton principle. The spanwise distributed PZT-4 sensor/actuator pair is embedded into the orthotropic host. The net voltage output from sensor is fed to a controller for the purpose of actuation. For piezoelectric actuation, active feedback control law and linear quadratic Gaussian controller are implemented. The purpose of present study is to investigate the validity of piezoelectric actuation under extreme conditions especially in critical region.

2. Analytical Model and Equations of Motion

Consider the thin-walled structure in which piezoelectric patch is embedded as Figure 1. The length [figure omitted; refer to PDF] of the blade is along [figure omitted; refer to PDF] direction. The origin of the rotating axis system is located at the rigid root in which the blade beam is mounted. [figure omitted; refer to PDF] is the rotating speed; [figure omitted; refer to PDF] is the chord length; [figure omitted; refer to PDF] is the thickness of section; [figure omitted; refer to PDF] is the radius of curvature of the middle surface; [figure omitted; refer to PDF] is the twist angle of section; [figure omitted; refer to PDF] is the wind velocity. It is assumed that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The equation of the middle line of the closed section is as follows [2]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] chord length and [figure omitted; refer to PDF] changes from 0~2π .

Figure 1: Coordinate system and aerodynamics and structure.

[figure omitted; refer to PDF]

The configuration of the piezoelectric actuator is assumed to be distributed over the entire blade span. The characteristics of the host composite structure and piezoelectric structure are depicted as (a) transverse shear, warping restraint effect, and secondary warping as depicted in [10]; (b) the master structure consists of 6-layer CAS configuration; the actuator is 1 piezoelectric layer, with the height [figure omitted; refer to PDF] and width [figure omitted; refer to PDF] along the circumferential [figure omitted; refer to PDF] , spanwise [figure omitted; refer to PDF] , and transverse [figure omitted; refer to PDF] directions; the width of piezoelectric layer [figure omitted; refer to PDF] is relatively small, generally less than or equal to [figure omitted; refer to PDF] chord length; (c) the piezoelectric elements may be employed concurrently for sensing and actuation and relate the voltage produced by the piezoelectric layer to the strain in the host structure; (d) the feedback control is achieved through the action of piezoelectrically induced vertical transverse shear motion at the blade tip.

To simplify the analysis, a structural tailoring technology for symmetric thin-walled structure is applied [10]. Consider the cantilever vibration first, ignore rotation about [figure omitted; refer to PDF] , and retain rotation about [figure omitted; refer to PDF] according to the requirements of piezoelectric actuation, the displacements [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are displayed in Figure 1, and the system kinetic energy [figure omitted; refer to PDF] and potential energy [figure omitted; refer to PDF] are, respectively, expressed as [figure omitted; refer to PDF] where the primary warping function [figure omitted; refer to PDF] is depicted as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the section area bounded by the midline and [figure omitted; refer to PDF] denotes the length of the contour midline. The radius of curvature of the middle surface is expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the stress component and represents the three-dimensional constitutive equations for the actuator layers with strain components [figure omitted; refer to PDF] . It can be deduced as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] means elastic coefficient and the last terms identify the actuation stresses induced by the applied electric field. In addition, [figure omitted; refer to PDF] is a spatial function expressed by Heaviside distribution [10]. The composite parameters and piezoelectric properties can be found in Table 1.

Table 1: Composite parameters and piezoelectric properties.

Composite items	Values	Piezoelectric items	Values
Maximum exterior width	24.21 × 10-3 m	Thickness [figure omitted; refer to PDF]	1.9 × 10-4 m
Maximum exterior height	2.643 × 10-3 m	Piezoelectric coefficient [figure omitted; refer to PDF]	-2.05 × 102 Pa V
Blade density [figure omitted; refer to PDF]	1672 kg/m3	Density	7.65 × 102 kg s2 /m4
Ply layers	6	Electrical permittivity [figure omitted; refer to PDF]	1.2 × 10-8 F/m
Ply thickness	127 × 10-6 m	Elastic coefficient [figure omitted; refer to PDF]	1.39 × 1011 Pa
Ply layers	6
[figure omitted; refer to PDF] / [figure omitted; refer to PDF]	3.5 GPa/0.34	Elastic coefficient [figure omitted; refer to PDF]	7.778 × 1010 Pa
[figure omitted; refer to PDF] / [figure omitted; refer to PDF]	25.8 GPa/8.7 GPa	Piezoelectric width [figure omitted; refer to PDF]	c/6 m

Based on the extended Hamilton principle, the governing system of cantilever vibration can be derived as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the virtual work of the nonconservative forces depicted as [10] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the piezoelectrically induced terms expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the applied electric field on which the piezoelectrically induced moment depends related with patch area [figure omitted; refer to PDF] . It is computed as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote local stretching and stretching-bending coupling rigidity quantities, respectively.

It is complicated to directly deduce the equations of motion of the rotating section with piezoelectric fiber rein forced composite actuators. Hence another simplified method is applied here. Based on (6), the equations of motion of cantilever vibration can be obtained first. With the rotating centrifugal motion [11] and the CAS structural tailoring technology being considered [10], the equations of rotating motion with vertical bending-twist-transverse shear coupling can be deduced as [figure omitted; refer to PDF] where the related composite structural parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are displayed in appendix, and the classical aerodynamics of lift [figure omitted; refer to PDF] and moment [figure omitted; refer to PDF] can be expressed as [2] [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

3. Solution Methodology

3.1. Application of Galerkin Method

To analyze the aeroelastic system given by (11a)-(11c), the Galerkin method is used so as to reduce the aeroelastic coupled equations into the state-space form [11]. Firstly the representation of displacement functions is as follows: [figure omitted; refer to PDF] where test functions are required to satisfy the kinematics and force boundary conditions of the cantilever blade and can be written as [figure omitted; refer to PDF]

Secondly substituting (15) into (11a)-(11c), with Galerkin method applied, gives [figure omitted; refer to PDF] matrix equations as follows: [figure omitted; refer to PDF] where the state variable [figure omitted; refer to PDF] , and the related coefficient matrices are [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Equation (17) is the linear decoupling equation governing the motion of the aeroelastic system, which is a matrix equation with [figure omitted; refer to PDF] subequation structure.

3.2. Active Control

In active flutter suppression, it is essential to properly select actuators and properly use control method and mathematical algorithm. The feedback control is achieved through the piezoelectrically induced vertical transverse shear motion at the blade tip, considered in conjunction with the implementation of a combined feedback control law, when external voltage of opposite sign is applied in the upper and bottom piezoactuator layer. The applied electric field [figure omitted; refer to PDF] on which the piezoelectrically induced moment depends may be expressed through a prescribed linear functional relationship with the kinematical response quantities characterizing the blade's response. The piezoelectrically induced bending moment [figure omitted; refer to PDF] intervenes solely in the boundary conditions prescribed at the blade tip and plays the role of a boundary moment control due to special distribution of piezoactuators [12]. A feedback control law is implemented with [figure omitted; refer to PDF] at the blade tip expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the velocity and acceleration feedback gains, respectively. So for active control with feedback law, system equation (17) is rewritten as [figure omitted; refer to PDF] where the new coefficient matrices are as follows: [figure omitted; refer to PDF]

In order to determine the time domain response of the dynamic system of (21), a Runge-Kutta time-marching approach is applied. Using the method presented in [1], (21) will be expressed in state-space form. In general, upon defining the state vector [figure omitted; refer to PDF] and adjoining the identity equation [figure omitted; refer to PDF] , (21) can be converted to the state-space expression: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] state matrix [figure omitted; refer to PDF] and [figure omitted; refer to PDF] matrix [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] Herein [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] unitary matrix, [figure omitted; refer to PDF] is [figure omitted; refer to PDF] unitary matrix, and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] zero matrix.

In order to minimize settling time and required control energy, optimal control techniques are necessary for controller design. Since these sensitive structures often experience time-varying loads, their safe and effective design requires accurate response [13]. The linear quadratic regulator (LQR) is often used to analyze vibration control of rotating composite beam. An advantage of the quadratic optimal control method is that it provides a systematic way of computing the state feedback control gain matrix [14]. In [13], the Galerkin method, along with either instantaneous or classical LQR methods, is used to analyze vibration control of a rotating composite pretwisted single-celled box beam, exhibiting transverse shear flexibility and restrained warping.

Here in order to suppress the too large initial vibration amplitude in LQR process and decrease the influence of measurement noise (might be produced by unsteady aerodynamics), the linear quadratic Gaussian (LQG) controller is used [4, 15]. It is assumed that the measurement noise and disturbance signals (process noise) are stochastic with known statistical properties and are hidden in system equation (23). The dynamic equation of LQG controller can be represented as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the weighting matrices of system outputs and control inputs, respectively; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the gain matrices of the Kalman state estimator and the gain matrix of the optimal regulator, respectively; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the positive finite solution of algebraic Riccati equations as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the intensity matrices of the gust input and the Gaussian white noise process of measurement, respectively. For the LQG control law, the gain matrix [figure omitted; refer to PDF] can be determined firstly by choosing the weighting matrices as [figure omitted; refer to PDF] being 6 [figure omitted; refer to PDF] unitary matrix and [figure omitted; refer to PDF] . Then the gain matrix [figure omitted; refer to PDF] of the Kalman state estimator is determined via [figure omitted; refer to PDF] and [figure omitted; refer to PDF] being [figure omitted; refer to PDF] unitary matrix.

The LQG control problem is to find the optimal control [figure omitted; refer to PDF] which minimizes [figure omitted; refer to PDF] and the required solution to the LQG problem is then found by replacing [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , to give the expression [figure omitted; refer to PDF]

4. Results and Discussion

To testify the validity of active flutter suppression, numerical results of time responses at blade tip for LQG controllers are presented, compared with related results of other cases. Generally, rotating blades have specific rotating speed, ply angle, and tip speed ratio for operation. Here basic testing parameters are ply angle [figure omitted; refer to PDF] ; tip speed ratio of wind turbine [figure omitted; refer to PDF] ; blade length [figure omitted; refer to PDF] m.

In general, the critical wind speed of a 2D section can be determined by v-g method [4] by eigenvalue analysis. However, the object of present research is a 3D entity with bending-twist-transverse shear coupling. After decoupling the number of subequations in (17) reaches [figure omitted; refer to PDF] (here [figure omitted; refer to PDF] ), with the number of eigenvalues being [figure omitted; refer to PDF] . V-g method has lost the utility. However, we can use another approximate method by the eigenvalue analysis. For different wind speeds, the maximum value of all the eigenvalues of each wind speed is determined, and then the critical wind speed is determined from the fluctuation of the largest eigenvalues. Figure 2(a) displays maximum real parts of eigenvalues versus velocities from 20~45 m/s. It can be obviously demonstrated that the critical wind speed is [figure omitted; refer to PDF] m/s.

Figure 2: Maximum real parts of eigenvalues versus velocities (a) and the responses for both [figure omitted; refer to PDF] m/s and [figure omitted; refer to PDF] m/s under situation of without piezoelectric materials (b).

(a) Maximum real parts of eigenvalues versus velocities

[figure omitted; refer to PDF]

(b) The responses of vertical bending [figure omitted; refer to PDF] , twist [figure omitted; refer to PDF] , and transverse shear [figure omitted; refer to PDF] motions for both [figure omitted; refer to PDF] m/s and [figure omitted; refer to PDF] m/s

[figure omitted; refer to PDF]

For flutter analysis, an approximate critical state range based on [figure omitted; refer to PDF] m/s~34.695 m/s is considered. When [figure omitted; refer to PDF] m/s, the system is convergent with smaller amplitude. When [figure omitted; refer to PDF] m/s, the system presents the divergent state with a large amplitude.

In general, the damping ratio increases constantly as the feedback control gain increases. However feedback control gain cannot be increased infinitely because the applied voltage must be limited for the sake of breakdown voltage of actuators [16]. According to the requirements of blade structure and actual control hardware here, the fixed feedback gains of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are applied.

Figure 2(b) shows the time responses of vertical bending [figure omitted; refer to PDF] , twist [figure omitted; refer to PDF] , and transverse shear [figure omitted; refer to PDF] motions under situation of without piezoelectric materials. Two critical wind speed values are considered at the same time; one is [figure omitted; refer to PDF] m/s (in blue mark) and the other one is [figure omitted; refer to PDF] m/s (in green mark).

From the whole response trend in the case of [figure omitted; refer to PDF] m/s, the three displacements seem to be convergent. In fact for the vertical bending displacement [figure omitted; refer to PDF] , within 2 s time, the amplitude of the vibration quickly exceeds the length of the blade [figure omitted; refer to PDF] , so actually the vertical bending [figure omitted; refer to PDF] motion has been in the state of divergence, so do twist [figure omitted; refer to PDF] and transverse shear [figure omitted; refer to PDF] motions. The instability also can be testified by Imag.-Real plot of eigenvalues of homogeneous equation system in Figure 3(a), where some closed-loop poles lie in the right-half plane, resulting in an unstable system.

Figure 3: Imag.-Real plot of homogeneous equation system for [figure omitted; refer to PDF] m/s defined in state-space.

(a) Without piezoelectric materials

[figure omitted; refer to PDF]

(b) With gains of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , but without optimal controller

[figure omitted; refer to PDF]

(c) With LQG controller

[figure omitted; refer to PDF]

As for another case of [figure omitted; refer to PDF] m/s in Figure 2(b), the three displacements rapidly diverge from the start with larger amplitude. These two cases in Figure 2(b) both can be regarded as the unstable states.

Take the case of [figure omitted; refer to PDF] m/s; for example, analysis and discussion for active control and flutter suppression will be carried out in the following in Figures 3~5.

Figure 3 shows Imag.-Real plot of homogeneous equation system (only concerning the characteristic matrices of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ) for [figure omitted; refer to PDF] m/s defined in state-space: (a) without piezoelectric materials; (b) with gains of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , but without optimal controller; (c) with LQG controller. Figures 3(a)~3(b) show the system instability. Some closed-loop poles lie in the right-half plane, and others are in the left-half plane. It can be seen from the Imag.-Real plot of Figure 3(c) that the homogeneous equation system obtained by control of LQG controller is stable, with all the closed-loop poles being in the left-half plane.

Figure 4 shows comparisons of the three displacements responses for [figure omitted; refer to PDF] m/s with fixed feedback gains of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , which are characterized by the state without optimal controller (in blue mark) and LQG controller (in red mark).

Figure 4: The responses of the three displacements under condition of fixed feedback gains of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , which are characterized by the state without optimal controller (in blue mark) and LQG controller (in red mark) for [figure omitted; refer to PDF] m/s.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Figure 5: The time responses of the three displacements for wind velocity of [figure omitted; refer to PDF] m/s, which are characterized by the state without piezoelectric materials and actuation by LQG controller.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

It can be seen that piezoelectric actuation based on fixed feedback gains of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and without optimal controller (blue mark in Figure 4) can greatly suppress the flutter, compared with the results in Figure 2(b). However, the results of these three displacements still maintain a larger vibration amplitude, and what is more, combined with the eigenvalue analysis in Figure 3(b), the system (with blue mark in Figure 4) might be conditionally stable. In practice, conditionally stable systems are not desirable. If system parameter assumes a value corresponding to unstable operation, the system may break down or may become nonlinear due to a saturation nonlinearity that may exist. Therefore, further active control based on theoretical algorithm, such as LQG algorithm, is necessary.

As far as the trend of response is concerned, the three displacements based on LQG controller in Figure 4 are rapidly convergent from the start. Figure 4 (red mark) also shows that the flutter amplitudes of all the three displacements decrease rapidly with the change of time and tend to be steady within 2 s. It shows obvious effect of LQG flutter suppression on aeroelastic instability.

In contradistinction with the trends without feedback control occurring in Figure 2(b) for [figure omitted; refer to PDF] m/s, the results displayed in Figure 4 reveal that LQG algorithm and active feedback control law have positive effects on classical flutter suppression. In order to verify the LQG law of universal, a further validation of LQG controller concerns the case of rapidly divergent state characterized by wind velocity of [figure omitted; refer to PDF] m/s without piezoelectric materials in Figure 5 (blue mark). From this figure it becomes apparent that when the wind speed is greater than the critical value, with increase of time (within 1 s), the magnitudes of the vibration amplitudes of the three displacements increase rapidly, even to unbelievable extent.

The effect of LQG controller on flutter suppression is illustrated in Figure 5 (green mark). The results show an expected conclusion, namely, that, for wind speed beyond the critical value, even in the adverse circumstances, the effect of LQG controller on flutter suppression is prominent. As a matter of fact, all the three responses under LQG controller are always stabilized to a constant value with smaller deviation when the time is extended beyond 2 s; meantime the three velocities at the blade tip are stabilized to zero within 2 s (see Figure 6), which verify the control performance for the state variables of (29).

Figure 6: The three velocities at the blade tip.

[figure omitted; refer to PDF]

It should be stated that the purpose of present study is to investigate the validity of piezoelectric actuation under extreme conditions for [figure omitted; refer to PDF] m/s as mentioned in Figures 2(b)~6. Of course, in general, the LQG algorithm is effective. A further validation of LQG controller concerns that the cases of low velocity for [figure omitted; refer to PDF] m/s and [figure omitted; refer to PDF] m/s are displayed in Figure 7. The results show the same conclusion as Figure 5, namely, that, for wind speed in low area, the effect of LQG controller on flutter suppression is obvious. Simultaneously all the three responses under LQG controller are always stabilized to constant values when the time is extended beyond 3 s.

Figure 7: The time responses of the three displacements for wind velocities of [figure omitted; refer to PDF] m/s (a) and [figure omitted; refer to PDF] m/s (b), which are characterized by the state without piezoelectric materials and actuation by LQG controller.

(a) [figure omitted; refer to PDF] m/s

[figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF] m/s

[figure omitted; refer to PDF]

In addition, the cost of piezoelectric materials is high with the mass of the system increasing as more actuator is used, with greater length and thickness being applied. Hence, adequate flutter suppression performance with LQG controller might be obtained through the proper arrangement and distributed size of sensor/actuator pair.

5. Conclusions

The purpose of present research is to investigate the validity of piezoelectric actuation under extreme conditions. An analytical study devoted to the mathematical modeling of single-cell thin-walled composite wind turbine blade beam is presented in the paper. Simultaneously classical flutter and active control based on piezoelectric actuation are investigated. The validity of the piezoelectric actuation is tested and illustrated by time domain response analysis. Since the related research on wind turbine is scarce, only numerical simulation technology is investigated here. The numerical illustrations reveal validation of the solution methodology and control algorithm used in the paper. Some concluding remarks can be drawn from the results as follows:

(1) Flutter suppression for classical flutter blade with vertical bending-twist-transverse shear coupling is investigated and discussed. Galerkin method is used to solve the coupling aeroelastic equations.

(2) Piezoelectric actuation is realized by active controller with feedback gains. It is obviously demonstrated that the LQG controller is robust and the effect on flutter suppression is apparent.

(3) It should be stated that the performance of LQG regulation depends on the choice of weighting matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] which have no analytic solution. Hence in some sense, the LQG controller in fact is man-made.

Acknowledgment

This work is supported by the Natural Science Foundation of Shandong Province of China (Grant ZR2013AM016).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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Appendix

Appendix

Structural Parameters and Rigidity Quantities

Consider [figure omitted; refer to PDF]