DESIGN OF MODEL-BASED FEEDFORWARD ... - CiteSeerX

DESIGN OF MODEL-BASED FEEDFORWARD COMPENSATORS FOR VIBRATION COMPENSATION IN A FLEXIBLE STRUCTURE

Max Rotunno Research & Development Aerostudi SpA

viale del Lido 37 Rome 00122, Italy Email: [email protected] Raymond A. de Callafon∗ University of California, San Diego

Dept. of Mechanical and Aerospace Engineering 9500 Gilman Drive La Jolla, CA 92093-0411, U.S.A. Email: [email protected]

ABSTRACT

This paper presents a systematic control relevant approach for the design of a feedforward compensation to control the effects of external vibration disturbances on a flexible structure. The design philosophy of the feedforward compensation is to reduce the flexibilities of the structure and thereby reducing the negative structural effects of an external vibration. This is done by computing a feedforward compensation via an optimal control scheme which allows for a trade-off between control energy and disturbance rejection. Additionally, the design approach is able to take into account possible mechanical coupling between the feedforward compensation and the measurement of the external disturbance. The approach is illustrated successfully by an experimental study based on a flexible structure equipped with an active suspended mass system for vibration control. 1

Introduction

Controlled reduction of the effect of external disturbances is crucial for flexible mechanical systems that are subjected to structural safety and system performance requirements. Passive dynamic vibration absorbers (DVA) such as tuned mass dampers (TMD) [1–4] and schemes based on seismic isolation [5–7] have been proposed to reduce vibrations in a flexible structure. ∗ Corresponding

author. Phone: +1.858-3444265; Fax: +1.858-8223107

1

Passive DVA are inherently 2nd or 4th order mechanical systems and allow to be tuned for only one or two frequencies of the structure [4, 8]. By using an adjustable TMD system, this design restriction can be avoided but requires the design of a TMD system with an tunable damper or spring configuration [9]. An active mass damper (AMD) system (that is not necessarily tuned) is more versatile and uses an additional forcing function on the mass to perform a broad range of disturbance attenuation by feedforward, feedback or a hybrid form of control [10]. In case external vibration disturbances can be measured reliably, an AMD system with feedforward control can provide effective disturbance rejection [9]. The design freedom in feedforward control lies in the dynamic filter used for the compensation of the external disturbance. Adaptive algorithms for the filter design, mainly based upon variations and extensions of the least mean squares algorithm [11–13], focus on stability and convergence issues. More importantly for a proper design of a feedforward controller is the ability to control the amount of control energy transmitted into the structure [14] and to take into account the possible mechanical coupling between the AMD system and the measurement of the external disturbance. Both phenomena are important in a robust AMD design for a lightweight flexible structure, where the AMD influences the coherent measurement of the vibration source [9]. In this paper, an AMD system is used to reduce the negative effects of an external disturbance on a flexible structure. The novelty of the external disturbance attenuation presented in this paper lies in the actual design of a fixed-gain feedforward compensation. First of all, the feedforward compensation is computed via an optimal control design which allows for a systematic trade-off between control energy and disturbance rejection. Secondly, the design approach is able to take into account possible mechanical coupling between the AMD system and the measurement of the external disturbance. The paper is organized as follows. The structure and the design of the proposed feedforward disturbance compensation is presented in Section 2. In Section 3, the modeling procedure for the design of the feedforward compensation is discussed in more detail. The experimental study to illustrate the effectiveness of the proposed feedforward design is reported in Section 4. Concluding remarks of the paper are summarized in Section 5. 2 Active disturbance compensation 2.1 Feedforward with mechanical coupling

In case external vibration disturbances can be measured coherently, feedforward control of an Active (tuned) Mass Damper (AMD) system can provide effective disturbance rejection. Crucial for the design of a feedforward controlled AMD system is the ability to model the flexible structure and to influence the amount of control energy transmitted into the structure. The control energy may be limited in practical applications where excessive control of the AMD leads to actuator saturation. Alternatively, the control energy can influence the coherent measurement of the external vibration used for the feedforward compensation. Both aspects need to be addressed to formulate a robust design of an AMD design for a (lightweight) flexible structure. Consider the simplified representation of an AMD controlled flexible structure depicted in Fig. 1. For illustrative purposes, the flexible structure is represented by lumped masses mi with flexible interconnections. A force disturbance f is assumed to act on the (base of the) flexible structure. The disturbance is assumed to be measured via a (base) acceleration v. To compensate 2

for the external measurable disturbance v, the flexible structure is equipped with an AMD for which the actuation force is indicated by the signal u. The signals xi , i = 1, 2, . . . n denote the acceleration signals at n crucial locations of the structure. The signals xi are used to characterize the vibration performance of the structure. For notation and modeling purposes the dynamic relation between the various signals are described by transfer functions. To anticipate on the results of Section 3, these transfer functions are not derived from the lumped mass behavior indicated in Fig. 1, but will be derived more generally via system identification techniques. Only the measurable acceleration disturbance v is used to create a feedforward compensation F with u = Fv

(1)

to reduce the effects of the vibrations on the flexible structure. With Eq. (1), the control energy can be characterized by considering the 2-norm kuw k2 of the weighted control signal uw = Wu

(2)

where W is a frequency dependent weighting function to emphasize certain frequency components in the control signal u. The signal xi is influenced negatively by the external disturbance v according to xi = Hi v where Hi is a dynamic model that relates the disturbance signal v and the acceleration at location i. Similarly, the AMD influences xi via x i = Gi u where Gi is a dynamic model that relates the control signal u and the acceleration at location i. In general, the AMD will also influence the acceleration measurement v via v = Gc u

(3)

where Gc indicates the mechanical coupling between AMD excitation and the coherent measurement of the external vibration. Lumping the signals xi in one performance signal e, the effect of the feedforward compensation in Eq. (1) and the coupling in Eq. (3) can be seen clearly in the block diagram of Fig. 2, where the single input multi-output transfer function H and G are composed of the transfer functions Hi and Gi respectively. 3

As often seen in active feedforward control of sound and vibration [7], the feedforward controller F and the coupling Gc create a feedforward compensation u = F(I − Gc F)−1 d

(4)

and reflects a positive feedback connection of Gc and F. The presence of a mechanical (or acoustic) coupling Gc might easily lead to an undesirable or even unstable feedforward compensation F(I − Gc F) if Gc it is not taken into account in the design of F. In the design of the feedforward compensation F, both the positive feedback connection in Eq. (4) and requirements on the allowable control energy given by Eq. (2) have to be taken into account. 2.2

Hybrid design of optimal feedforward

To postulate the design of a fixed-gain feedforward controller, we adopt an optimal control design strategy that exploits the positive feedback of the feedforward compensator F and the mechanical coupling Gc as depicted in Fig. 2. The optimal control design approach is similar to the standard plant methodology presented in Clark and Bernstein [15] and Par´e and How [10] but the methodology in this paper systematically makes a trade-off between control energy formalized in Eq. (2), disturbance rejection and mechanical coupling. This is achieved by adding appropriate weighting functions Wd and Wu to the block diagram shown in Fig. 2 so as to produce the block diagram shown in Fig. 3. The weighting functions are used to specify the control objectives for the design of the feedforward controller. The role of the weighting functions in the systematic design can be explained as follows. 1. Wd is used to take into account the spectrum of the external disturbance d. 2. Wu is used to shape the input to the AMD to the desired value, so that energy is feed into the system only at those frequencies where it is necessary. Once the models H, G, Gc and the weighting functions Wd and Wu are given, the standard plant description depicted in Fig. 4 can be constructed from Fig. 3. The input vector w in Figure 4 reflects the external disturbance for which the effect on the performance signal e needs to be minimized. Referring to Figure 3, the signal w is passed through the weighting function Wd so as to generate the ground acceleration input d. The performance signal e is built up from the combination of the output error signal e and the filtered signal uw = Wu u. The signal y is the measured ground acceleration v that will act as an input to the feedforward controller and u is the input to the AMD system for control purposes. Using basic algebraic manipulations it can be shown that the generalized plant P is given by    HWd (HGc + G) · ¸ e1  w  e2  =  0 Wu u y Wd −Gc 

(5)

Once the the generalized plant has been formulated, a feedforward controller can be computed via a standard H2 optimization technique [16]. 4

3

Modeling for feedforward design

To be able to use the feedforward control design procedure discussed above, it is necessary to have mathematical models of the transfer function H, G and Gc . These models can be obtained either by first principles modeling or, as will be done in this paper, by system identification techniques. In such techniques, time domain data from dedicated experiments is used to determine a frequency response function. A dynamical model is then obtained by curve fitting the frequency response data. 3.1

Frequency Response Estimation

Let u(t), t = 1, . . . , N indicate the discrete time signal inputted into a system and y(t) the system response due to the excitation. The discrete Fourier transforms of u(t) and y(t) are given by: 1 N UN (ω) = √ ∑ u(t) exp−iωt N t=1

(6)

1 N YN (ω) = √ ∑ y(t) exp−iωt N t=1

(7)

and the empirical transfer function estimate (ETFE) is computed by Ljung [17] YN (ω) Tˆ (eiω ) = UN (ω)

(8)

To reduce the effects of measurement noise and the variance of the ETFE, the estimate given in Eq. (8) is averaged over M measurements, Tˆ (eiω ) =

M

∑ Tˆk (eiω)

k=1

With respect to th notation used in Section 2, a ETFE of Hi can be obtained by measuring the base acceleration and the acceleration signal xi of interest and applying the procedure above so as to obtain a Hˆ i . In a similar way it is possible to determine Gˆ i and Gˆ c . 3.2

Control relevant modeling

Given an estimated frequency response data Tˆ ( jωk ) (where Tˆ is used to represent either Hˆ i , Gˆ i or Gˆ c ) along the frequency grid Ω of length N, the aim of the frequency domain identification is to find a (discrete time) linear time invariant model M of limited complexity that approximates the data. These models can then be used to determine the natural frequencies and damping of the system and to design a digital feedforward controller. To address the limited complexity, the model M to be determined is parameterized in a transfer function representation M(z, θ) :=

b0 + b1 z−1 + · · · + bn z−n 1 + a1 z−1 + · · · + an z−n 5

(9)

where z = e jω denotes the z-transform variable and θ := [b0 , b1 , · · · , bn , a1 , · · · , an ] denotes a real valued parameter of unknown coefficients in the transfer function representation given in (9). Furthermore, it can be seen that the order or complexity of the linear model can be specified with the integer value n. The approximation of the data Tˆ ( jωk ) by the model M(z, θ) is addressed by considering the following curve fit error E( jωk ) := [Tˆ ( jωk ) − M(e jωk , θ)]W ( jωk )

∀ωk ∈ Ω

(10)

that needs to be minimized. In (10), W ( jωk ) denotes a scalar weighting function that is used to influence the curve fitting of the frequency response data. With the definition of the curve fit error (10), a parameter θˆ is estimated by solving the following non-linear weighted least squares minimization: N

θˆ = arg min ∑ E( jωk )E ∗ ( jωk ) θ k=1

where ∗ is used to denote the complex conjugate transpose. 4

Experimental study

An experimental setup has been developed at the University of California at San Diego’s (UCSD) System Identification and Control Laboratory (SICL). The experimental setup as shown in Figure 5 consists of a shaker table, and an AMD-equipped two-degree of freedom flexible structure. The experimental setup can be used to evaluate the performance of various control algorithms as applied to vibration-suppression of flexible structures. The AMD consists of a linear actuator attached to a mass spring setup as shown in Figure 5. The springs attached to the mass were chosen such that the actuator’s bandwidth is greater than the second resonant mode of the building. The shaker table provides a uni-axial disturbance input to the building and can replicate an arbitrary input motion signal with a maximum of 3g acceleration in a frequency range from DC to 10kHz. Two accelerometers with a bandwidth of 200Hz are attached to the structure. One accelerometer is located at the base of the shaker table to provide the acceleration of the base motion, while the second accelerometer is located in correspondence to the first mass of the structure. 4.1

Identification of dynamics

In the experimental setup described above, only the first mass acceleration signal is used to characterize the vibration performance of the flexible structure. The transfer function H is therefore a single input single output (SISO) system, which relates the base and first floor accelerometer signals. In the same way, the transfer function G is SISO and together with Gc they relate the input to the AMD with the first mass and base accelerometer signals respectively. 6

Below are the results obtained by applying the frequency response estimation and curve fitting procedures described in Section 3.2. The discrete time models H, G and Gc all have a sampling frequency of 200Hz and models of 4th and 6th order are used to capture the main resonance modes present in the structure. From Fig. 6 it can be seen that in the transfer function of H two main resonance modes are located at approximately 3.4Hz and 12Hz. The first mode at 3.4Hz is due to the first fundamental mode of the structure and the second resonance at 12Hz is due to the AMD. The transfer function of H also contains a zero at approximately 5.5Hz, which is due to a blocking property of the structure. With a sinusoidal base excitation of 5.5Hz, the mass of the first floor is hardly moving due to residual vibrations of the mass of the second floor. The second fundamental mode of the structure is highly damped and has a frequency of 8.5Hz. The effect is modeled in the transfer function of G and Gc depicted in Fig. 7 and Fig. 8. It can be noted that the AMD has most of its control capability in the 12Hz region. This was to be expected since this is also where the AMD has its natural frequency and the force required to produce a displacement of the AMD goes to zero. From the transfer function of the mechanical coupling Gc in Fig. 8 it can also be observed the coupling is small and is similar to the dynamics of G. The coupling should however be taken into account when designing the feedforward compensator F to avoid performance degradation of the vibration controller or even instabilities when implementing the feedforward compensator. 4.2

Design and implementation of feedforward

The objective of the feedforward control design is to be able to control the vibrations of the first mass of the flexible structure shown in Fig. 5. The signal e to be minimized in the H2 feedforward design procedure is therefore the acceleration of the first mass of the system. The weighting function Wd was chosen to be unity, since it is interesting to see if the AMD can accomplish a broadband vibration reduction. The choice of the weighting function Wu can be motivated in the following way: from the magnitude plot of the frequency response of H in Fig. 6, it is possible to see that the feedforward controller should input energy so as to cancel the resonance peaks of H and should not input energy where H has a zero. This specification can be accomplished by simply taking Wu to be the stable inverse of H as shown in Fig. 9. Fig. 10 shows the controller obtained by applying the H2 optimization procedure. As anticipated, most of the control energy is concentrated around the first vibration mode of the structure, since this is where the AMD is less effective. Figure 10 also shows the predicted reduction obtained in the transfer function between disturbance d and the output e. The feedforward controller obtained above was implemented on the experimental test bed. Figure 11 shows the spectrum of the output (acceleration of the first mass) with and without the feedforward controller. The reductions under the peaks is consistent with the theoretical predictions. Figure 12 shows the time domain response of the structure.

5

Conclusions

Measurements of the external disturbance can be used to reduce the effect of a disturbance by feedforward filtering or compensation. In vibration isolation the design philosophy of feedforward compensation is to reduce the flexibilities of the structure and thereby reducing the negative 7

structural effects of an external vibration. For a proper design of a feedforward filter, the dynamics of the disturbance, the feedforward actuator and possible coupling between the feedforward actuation and the measurement has to be taken into account. The design of the feedforward compensator for vibration cancellation presented in this paper takes into account the mechanical coupling between the feedforward compensation and the measurement of the external disturbance and poses the feedforward filter design as an optimal control design which allows for a trade-off between control energy and disturbance rejection. Weighting functions that address the trade-off between control energy and disturbance rejection are chosen on the basis of the disturbance dynamics. In this paper the feedforward filter design is solved by a standard H2 -optimal control problem, but the results can easily be extended to include H∞ control design methods for robust design of the feedforward filter. The approach is illustrated successfully by an experimental study based on a flexible structure equipped with an active suspended mass system for vibration control. It is shown that the feedforward controller is able to dampen out the main resonance modes due to disturbance excitation and reducing the vibration of the structure with an order of a magnitude without excessive control signals from the feedforward compensator. NOMENCLATURE

ai Denominator polynomial coefficients bi Numerator polynomial coefficients d External acceleration disturbance ei Output performance signals used in the control design E( jω) Frequency dependent curve fit error F Transfer function of feedforward compensator Gi Transfer function of control signal u to xi Gc Transfer function of mechanical coupling between u and v H Transfer function from v to single specific acceleration signal x to be minimized Hi Transfer function of external disturbance v to xi M(z, θ) Parametrized discrete time model for estimation purposes n Order of (discrete) time model to be estimated N Number of data points in time or frequency domain ω Frequency vector in rad/s Ω Frequency grid Tˆ ( jω) Emperical Transfer Function Estimate as a function of ω θ Free parameter in model estimation θˆ Estimated parameter of model u Controller output signal UN Discrete finite time Fourier transform of u uw Weighted controller output signal v Measurable external acceleration disturbance w Input performance signals used in the control design W ( jω) Frequency dependent weighting function Wd Transfer functions that models spectrum of d Wu Transfer functions for filtering of control signal 8

xi Acceleration signal at location i y Controller input signal YN Discrete finite time Fourier transform of y z z-transform variable REFERENCES

[1] P. Wirsching, G. Campbell, Minimal structural response under random excitation using the vibration absorber, Eartquake Engineering and Structural Dynamics 12 (1974) 303–312. [2] J. Sladek, R. Klinger, Effect of tuned-mass damper on seismic response, ASCE Journal of Structural Engineering 109 (1983) 2004–2009. [3] B. Korenev, L. Reznikov, Dynamic Vibration Absorbers: Theory and Technical Applications, John Wiley, NY, 1993. [4] R. Burdisso, J. Heilmann, A new dual-reaction mass dynamic vibration absorber actuator for active vibration control, Journal of Sound and Vibration 214 (5) (1998) 817–831. [5] M. Constantinous, I. Tadjbakhsh, Probabilistic optimal base isolation of structures, ASCE Journal of Structural Engineering 109 (1983) 676–689. [6] I. Buckle, R. Mayes, Seismic isolation: history application and performance - a world view, Earthquake Spectra 6 (1990) 161–201. [7] C. Fuller, S. Elliot, P. Nelson, Active Control of Vibration, Academic Press, London, UK, 1996. [8] D. Anthony, S. Elliot, A. Keane, Robustness of optimal design solutions to reduce vibration transmission in a lightweight 2D structure, part I: geometric redesign, Journal of Sound and Vibration. [9] D. Anthony, S. Elliot, Robustness of optimal design solutions to reduce vibration transmission in a lightweight 2D structure, part II: application of active vibration control techniques, Journal of Sound and Vibration 229 (3) (2000) 529–548. [10] T. Par´e, J. How, Hybrid H2 control design for vibration isolation, Journal of Sound and Vibration 226 (1) (1999) 25–29. [11] S. Elliot, J. Stothers, P. Nelson, A multiple error LMS algorithm and its application to the active control of sound and vibration, IEEE Trans. on Acoustics, Speech, and Signal Processing 35 (1987) 1423–1434. [12] J. Vipperman, R. Burdisso, C. Fuller, Active control of broadband structure vibration using the LMS adaptive algorithm, Journal of Sound and Vibration 166 (1993) 283–299. [13] R. Cabell, C. Fuller, A principal component algorithm for feedforward active noise and vibration control, Journal of Sound and Vibration 227 (1) (1999) 159–181. [14] C. Howard, S. Snyder, C. Handens, Calculation of vibration power transmission for use in active vibration control, Journal of Sound and Vibration 233 (4) (2000) 573–585. [15] R. Clark, D. Bernstein, Hybrid control: separation in design, Journal of Sound and Vibration 214 (4) (1998) 784–791. [16] K. Zhou, J. Doyle, Essentials of Robust Control, Prentice-Hall, 1998. [17] L. Ljung, System Identification, Prentice Hall, 1998.

9

uc

xn¾

mn

S ¶ S¶ ¶S ¶ S

xn−1 ¾

mn−1

x2¾ x1¾

m1

S ¶ S¶ ¶S ¶ S ¾ e e

v¾

Figure 1.

m2

S ¶ S¶ ¶S ¶ S

Flexible structure with AMD on top floor, acceleration signals

f

xi , i = 1, 2, . . . n and measurable base disturbance

acceleration v

d -+d v − 6 Gc

-

F

6

Figure 2.

u

-

G

Block diagram of feedforward controller AMD system

w - W d-+d v d − 6 Gc

-

+

-d +6

H

e1 2 - Wu e-

F

6

Figure 3.

-+d e+6

H

u

-

G

Feedforward block diagram with weighting functions

w-

eP y

u F

Figure 4.

¾

Generalized plant

10

Figure 5.

Experimental Setup

|H| magnitude

0

10

−1

10

−2

10

−1

10

0

1

10

10

2

10

phase [degree]

∠H 0

−50

−100

−150

−200 −1 10

0

1

10

10

2

10

frequency [Hz] Figure 6.

Bode plot of measured frequency response(dashed) and 4th order parametric model (solid)

11

|G| magnitude

0

10

−2

10

−4

10

−1

10

0

1

10

10

2

10

∠G phase [degree]

0

−200

−400

−600

−800 −1 10

0

1

10

10

2

10

frequency [Hz] Figure 7.

Bode plot of measured frequency response (dashed) and 6th order parametric model (solid)

|Gc |

0

magnitude

10

−1

10

−2

10

−3

10

−1

10

0

1

10

10

2

10

phase [degree]

∠Gc 0 −200 −400 −600 −800 −1 10

0

1

10

10

2

10

frequency [Hz] Figure 8.

Bode plot of measured frequency response (dashed) and 6th order parametric model (solid)

12

0

magnitude

10

−1

10

−2

10

0

1

10

10

2

10

frequency [Hz]

Amplitude Bode plot of control input weighting function Wu

Figure 9.

0

magnitude

10

−1

10

−2

10

0

1

10

10

2

10

frequency [Hz] Figure 10.

Amplitude Bode plot of Feedforward controller (dashed), open loop uncontrolled system (dotted) and controlled system

(solid) with feedforward controller and mechanical coupling

13

−2

10

−3

magnitude

10

−4

10

−5

10

−6

10

0

1

10

10

frequency [Hz] Figure 11.

Experimental verification of feedforward controlled AMD system: uncontrolled system (dotted) and controlled system

(solid)

0.8

0.6

acceleration [V]

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time [sec] Figure 12.

Time trace of acceleration signal to be minimized: open loop system (dotted) and feedforward controlled AMD system

with mechanical coupling (solid)

14