Multi-objective optimal design of active vibration ...

Journal of Sound and Vibration 339 (2015) 56–64

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Multi-objective optimal design of active vibration absorber with delayed feedback Rong-Hua Huan a, Long-Xiang Chen b, Jian-Qiao Sun c,n a b c

Department of Mechanics, Zhejiang University, Hangzhou 310027, China Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China School of Engineering, University of California, Merced, CA 95343, USA

a r t i c l e in f o

abstract

Article history: Received 5 January 2014 Received in revised form 2 November 2014 Accepted 12 November 2014 Handling Editor: D.J. Wagg Available online 15 December 2014

In this paper, a multi-objective optimal design of delayed feedback control of an actively tuned vibration absorber for a stochastically excited linear structure is investigated. The simple cell mapping (SCM) method is used to obtain solutions of the multi-objective optimization problem (MOP). The continuous time approximation (CTA) method is applied to analyze the delayed system. Stability is imposed as a constraint for MOP. Three conflicting objective functions including the peak frequency response, vibration energy of primary structure and control effort are considered. The Pareto set and Pareto front for the optimal feedback control design are presented for two examples. Numerical results have found that the Pareto optimal solutions provide effective delayed feedback control design. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Tuned vibration absorbers (TVAs) have been effectively used to remove undesirable oscillations from mechanical structures. The TVA usually consists of a spring–damper mass attached to the structure to be controlled [1]. However, the major drawback of the traditional TVA is that it is suitable only for a narrow bandwidth of operation. Therefore, it is useful in eliminating single frequency resonant vibrations. When the excitation frequency is unsteady or varying, the traditional absorber becomes ineffective and may potentially increase the base vibration. This paper presents a multi-objective optimal design of active TVAs with delayed feedback controls to overcome the shortcomings of the traditional TVAs. There have been many studies of active and semi-active TVAs [2–6]. One example of active TVAs is the well-known delayed resonator by Olgac and colleagues [7–12]. An important step in implementation of the delayed resonator is the determination of the feedback gain and delay time of the control. The delayed resonator can significantly eliminate the response of a primary system subject to a harmonic load of varying frequency. However, since the absorber works in a marginally stable state, there is a potential risk of instability for the system. An optimal delayed feedback vibration absorber is introduced in [13,14], which offers minimum peak frequency response within the given frequency range. The work in [15] explores the use of positive feedback proportional control using large time delays. The control parameters are chosen to maximize the stability margin. Renzulli et al. [16] have successfully demonstrated a robust novel automatic tuning DR in the experiments to suppress oscillations of a flexible beam. Proper design of feedback control parameters for active TVAs generally requires that a compromise between usually conflicting objectives. Examples include minimizing peak levels of the frequency response, flattening the frequency n

Corresponding author. E-mail address: [email protected] (J.-Q. Sun).

http://dx.doi.org/10.1016/j.jsv.2014.11.019 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

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response over a frequency range, and minimizing the control effort. In this paper, we consider such a multi-objective optimization problem (MOP) in designing active TVAs with delayed feedback. In particular, we consider a TVA attached to a primary discrete structure such as motors or engines subject to wideband random excitations. The simple cell mapping (SCM) method is applied to solve for the MOP [17–20]. The continuous time approximation (CTA) method [21,22] is used to obtain the response of the delayed system and to analyze the stability and statistics of the responses. The rest of the paper is organized as follows. In Section 2, we introduce the primary system with the attached TVA. In Section 3, we define the MOP and introduce the objective functions representing three conflicting objectives including the peak level of the frequency response, the vibration energy of primary structure over a frequency range and the control effort. In Section 4, we present two examples of active TVAs with delayed feedback. Section 5 concludes the paper. 2. Linear system with delayed vibration absorber Consider a linear system with delayed vibration absorber subject to random excitation acting on the primary mass #2 as shown in Fig. 1. The equations of motion are given by _ ¼ A0 X þ Aτ Xðt  τÞ þ GWðtÞ; X XðsÞ ¼ ϕðsÞ; s A ½  τ; 0; where X ¼ ½X 1 ; X 2 ; X_ 1 ; X_ 2 , and

2

0 6 6 0 6 A0 ¼ 6  k1 6 m1 4 k 1

m2

0 0

1 0

0 1

k1 m1

 mc11

c1 m1

 k1mþ2k2

c1 m2

 c1mþ2c2

0

0

0

0

0

0

 mkd1

0

kd m2

2

0 6 0 6 6 Aτ ¼ 6  kp 6 m1 4 k p

m2

(1) 3 7 7 7 7; 7 5

(2)

3

07 7 7 ; 07 7 5 0

(3)

  1 T G¼ 0 0 0 : m2

(4)

E½WðtÞWðt þ TÞ ¼ 2DδðTÞ:

(5)

W(t) is Gaussian white noise with zero mean and The power spectral density (PSD) of the white noise is a constant given by SWW ðωÞ ¼ D=π . For the linear system in Eq. (1), we can obtain an exact solution of PSD function for XðtÞ. Define a frequency response matrix as HðωÞ ¼ ½iωI A0  Aτ e  iωτ   1 :

(6)

The PSD function of the system is given by [23] SXX ðωÞ ¼ Hn ðωÞGSWW ðωÞGT HT ðωÞ:

(7)

Note that the original system lives in an infinite dimensional state space given by ðXðtÞ; Xðt  sÞ; 0 o s r τÞ. Applying the CTA method, we discretize the delayed part of the state vector by considering a mesh Ω ¼ fτi ; i ¼ 0; 1; …; Mg of M þ 1 points in ½0; τ such that 0 ¼ τ0 o τ1 o⋯ o τM  1 o τM ¼ τ. We define an extended state vector YðtÞ ¼ ½XðtÞ; Xðt  τ1 Þ; …; Xðt  τN ÞT  ½Y 1 ðtÞ; Y2 ðtÞ; …; Y M þ 1 ðtÞT :

(8)

Fig. 1. A two-degree of freedom linear system with delayed vibration absorber. Mass #2 represents the primary structure. Mass #1 is the absorber.

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After introducing an interpolation scheme of Xðt  τi Þ over the mesh grid YðtÞ without an explicit time delay to replace Eq. (1) [21,22]

τi, we obtain a stochastic equation for the vector

_ YðtÞ ¼ AYðtÞ þ BWðtÞ

(9)

The original four dimensional delayed system is converted into a high dimensional system without explicit time delay according to the CTA method. Then, following the same steps as for linear systems without time delay, the stability and the power spectral density function SYY ðωÞ of converted system (9) can be investigated. Obviously, SXX ðωÞ is a subset of the solutions contained in SYY ðωÞ. The accuracy of CTA as a function of the discretization level M and the ability of the method to predict the eigenvalues and time-domain responses are studied in [21,22]. Furthermore, the theoretical foundation of the method for predicting the rightmost eigenvalue of linear time-invariant systems with time delay is well documented in a book by Bellen and Zennaro [24]. The advantage of the CTA method lies in that it converts the root searching of the transcendental equation in the complex plane to an eigenvalue problem of a finite-dimensional system matrix. This feature is important to the optimization studies where the eigenvalue problems will be solved repeatedly for many times. 3. Multi-objective optimization problem The multi-objective optimization problem (MOP) for delayed vibration absorber design belongs to the following general class of problems. Consider the minimization of a vector-valued function: minfFðkÞg;

(10)

kAQ

where FðkÞ ¼ ½f 1 ðkÞ; …; f k ðkÞ; f i : Q -R 1 ;

F: Q -R k :

(11) q

fi are objective functions, k A Q is a q-dimensional vector of design parameters. The domain Q  R can in general be expressed in terms of inequality and equality constraints, Q ¼ fk A Rq j g i ðkÞ r0; i ¼ 1; …; l; and hj ðkÞ ¼ 0; j ¼ 1; …; mg:

(12)

Next, we define optimal solutions of the MOP by using the concept of dominance [25] described in the following set of definitions. (a) Let V; W A R k . The vector V is said to be less than W (in short: V o p W), if V i o W i for all i A f1; …; kg. The relation r p is defined analogously. (b) A vector v A Q is called dominated by another vector w A Q (w ! v) with respect to the MOP (10) if FðwÞ r p FðvÞ and FðwÞ a FðvÞ, otherwise v is called non-dominated by w. Note that if a vector w dominates a vector v, then w can be considered as a ‘better’ solution of the MOP. The definition of optimality or the ‘best’ solution of the MOP is now straightforward. (c) A point w A Q is called Pareto optimal or a Pareto point of the MOP (10) if there is no v A Q which dominates w. (d) The set of all Pareto optimal solutions is called the Pareto set denoted as P≔fw A Q : w is a Pareto point of the MOP ð10Þg:

(13)

The image FðPÞ of P is called the Pareto front. Recent studies seem to suggest that Pareto front may have fine structures for MOPs of control systems [20,26]. 3.1. Cell mapping method for MOP In this paper, we shall apply the simple cell mapping (SCM) method to obtain the Pareto optimal solutions for the delayed feedback control design [17–20]. The search of the Pareto optimal solutions is done in the discretized parameter space in a two step manner. In the first step, the design space is divided by an integer q-tuplet N ¼ ½N 1 ; N2 ; …; N q . An approximate Pareto set is found and then subdivided into smaller cells in the second step. The refined Pareto set is then obtained in terms of much smaller cells. For more details of the SCM method for MOP, the reader is referred to the references cited above. 3.2. Objective functions In this study, we have design parameters as k ¼ ½kp ; kd ; τ;

(14)

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Fig. 2. The power spectral density (PSD) SX 2 X 2 ðωÞ of the response of the primary system with a passive TVA.

and the design space Q ¼ fk A R3 jReðλðkÞmax Þ o  εg;

(15)

where λðkÞmax represents the largest real part of the eigenvalues of the matrix A of the converted system (9) and ε 40 is a small positive number to provide the robustness of stability. This robustness consideration differentiates the present work from the delayed resonator. In the following, we consider three objective functions. The first objective function consists of the oscillation energy of the stochastically excited primary structure in a relative wide frequency range. We take the integration of the power spectral density (PSD) function SX 2 X 2 ðωÞ of the displacement response of the primary structure within the frequency range ½ω1 ; ω2 , Z ω2 f 1 ðkÞ: S XX ¼ SX 2 X 2 ðωÞ dω; (16) ω1

where ω1 and ω2 denote the frequencies corresponding to the two resonant peaks of the PSD function as shown in Fig. 2. The second objective function represents the input energy of the delayed feedback control given by Z 1 Suu ðωÞ dω; (17) f 2 ðkÞ: S uu ¼ 0

where Suu ðωÞ is the power spectral density function of the delayed feedback control, which can be obtained from the power spectral density function SYY ðωÞ. The third objective function is the peak value of the PSD function of the displacement response of the primary structure, f 3 ðkÞ: Smax XX ¼ max SX 2 X 2 ðωÞ: ω A R1

(18)

This function value usually occurs at either ω1 or ω2 of the PSD function as shown in Fig. 2. We should point out that from the physics point of view, S XX would be in conflict with S uu and Smax XX , while the relationship between S uu and Smax XX can be more complicated. In summary, the above multi-objective optimization formulation provides far more freedom and flexibility to design vibration absorbers to meet different and sometimes conflicting goals. Unfortunately, most multi-objective optimization problems do not lead to analytic solutions even for simple linear systems. Development of efficient numerical algorithms is an important topic in the optimization community, and is out of the scope of this paper. 4. Numerical results 4.1. Position control We consider only the proportional position delayed feedback when kd ¼ 0. The design space for the parameters k ¼ ½kp ; τT is chosen as follows: Q ¼ fk A R2 jReðλðkÞÞmax o  ε; kp A ½ 3; 3; τ A ½0:1; 1:5g:

(19)

The parameters of the systems used throughout the paper are m1 ¼ 0:8, m2 ¼ 2:0, c1 ¼ 0:05, c2 ¼ 0:1, k1 ¼ 20:0, k2 ¼ 100:0, D¼ 1.57 and ε ¼ 0:01. The discretization number of the time delay is taken to be M¼150. Initially, we select the number of divisions in the two design parameters as N ¼ ½30; 30. The subsequent subdivision is ½7; 7. The Pareto set in the design space ½kp ; τ and the corresponding Pareto front in the objective space spanned by ½S XX ; S uu ; Smax XX  are shown in Figs. 3 and 4. Fig. 3 shows that along the delay parameter space, the optimal solutions can be separated into three separate parts. The first part is around the region Ω1 ¼ fτjτ A ½0:2; 0:4g, and is the largest one. Two smaller parts are around Ω2 ¼ fτjτ A ½0:6; 0:8g and Ω3 ¼ fτjτ A ½1:1; 1:3g. This result suggests that the delay time τ is better to be selected in Ω1 where the gain for the P control is negative, because the large Pareto set offers performance robustness to uncertainties in the system. Note that in Ω3, the gain for the P control is positive.

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Fig. 3. The Pareto set in the ½kp ; τ design space for the delayed feedback P control.

Fig. 4. The Pareto front in the objective space ½S XX ; S uu ; Smax XX  for the delayed feedback P control.

Fig. 5. The maximum value ReðλÞmax of the real part of eigenvalues of the matrix A as functions of delay time τ for different kp. (a) kp varying from  3 to  1; (b) kp varying from 1 to 3. Note that this is not the Pareto optimal solution. The figure illustrates the variation of stability with the time delay for fixed control gains.

The maximum value of the real part of eigenvalues of the matrix A as a function of delay time τ is shown in Fig. 5. There are four minima in the figure, marked as I, II, III and IV. The minima I and IV are located in the domain Ω1 and Ω3, respectively, and minima II and III are in Ω2. These minima are associated with the Pareto optimal solutions, which explain the disjoint Pareto optimal sets in the design space. Furthermore, we point out that Fig. 5 indicates the minimum I to be much more stable than the other three minima. This supports the early assessment that the better design resides in Ω1. The response PSD of the primary structure with and without optimal delayed feedback P controls is shown in Fig. 6. The PSD curves are flattened and the peak values of the PSD are reduced significantly for different Pareto optimal feedback controls. The time histories of the response of the primary structure with and without optimal delayed P feedback control are shown in Fig. 7.

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Fig. 6. The response PSD of the primary system with and without the delayed feedback P control for three Pareto optimal parameters ½kp ; τ: k1 ¼ ½  2; 98; 0:226; k2 ¼ ½  1:82; 0:235; k3 ¼ ½2:06; 1:27.

Fig. 7. The response of the primary system with and without the delayed feedback P control for different Pareto optimal parameters ½kp ; τ: k1 ¼ ½  2; 98; 0:226; k2 ¼ ½  1:82; 0:235.

Fig. 8. The Pareto set in the design space ½kp ; kd ; τ for the delayed feedback PD control.

4.2. PD control Now, we consider the PD delayed feedback control. The space for the design parameters k ¼ ½kp;kd ; τT is chosen as follows:

Ω ¼ fk A R3 j ReðλðkÞÞmax o  ε; kp ; kd A ½  3; 3; τ A ½0:1; 1:5g:

(20)

Initially, we select the number of divisions of the design parameter domain as N ¼ ½15; 15; 15. The subdivision is ½3; 3; 3. The Pareto set in the parameter space ½kp;kd ; τ and the corresponding Pareto front in the objective function space ½S XX ; S uu ; Smax XX 

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are shown in Figs. 8 and 9. The Pareto set consists of two clusters in the design space. One is around the region Ω1 ¼ fτjτ A ½0:1; 0:6g, and is bigger. The other cluster is smaller around the region Ω2 ¼ fτjτ A ½1:2; 1:5g. From Fig. 8, when the delay time τ is selected in Ω1, kp and kd are more likely to be negative. kp and kd are positive when τ is in Ω2. The maximum value of the real part of eigenvalues of the matrix A as a function of the delay time τ for the PD control is shown in Fig. 10. Similar to the previous example, the minima in Fig. 10 can explain the existence of the disjoint Pareto set.

Fig. 9. The Pareto front in the objective space ½S XX ; S uu ; Smax XX  for the delayed feedback PD control.

Fig. 10. The maximum value ReðλÞmax of the real part of eigenvalues of the matrix A as functions of delay time τ for different kp and kd : (a) kd ¼  0:5; kp varying from  3 to 3; (b) kd ¼ 2:0; kp varying from  3 to 3.0. Note that this is not the Pareto optimal solution. The figure illustrates the variation of stability with the time delay for fixed control gains.

Fig. 11. The response PSD of the primary system with and without the delayed feedback PD control for three Pareto optimal parameters ½kp ; kd ; τ: k1 ¼ ½  2; 93;  2:8; 0:46, k2 ¼ ½  2; 93; 0:93; 0:12, k3 ¼ ½0:67; 0:8; 1:48.

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Fig. 12. The response of the primary system with and without the delayed feedback PD control for different Pareto optimal parameters ½kp ; kd ; τ: k1 ¼ ½  2; 93;  2:8; 0:46, k2 ¼ ½  2; 93; 0:93; 0:12.

We should point out that in this example, some minima of the real part of eigenvalues are positive, indicating the instability. These minima are not part of the Pareto optimal solutions. The response PSD of the primary structure with and without the optimal delayed feedback PD controls is shown in Fig. 11. The peak values of the PSD curves are reduced significantly for different Pareto optimal solutions. Especially, for k ¼ ½  2; 93;  2:8; 0:46, the PSD curve is flattened almost to a horizontal line, which implies a good effectiveness of the optimally designed delayed vibration absorber over a wide band of frequencies. The time histories of the response of the primary structure with and without optimal delayed PD feedback control are shown in Fig. 12.

5. Concluding remarks We have presented a multi-objective optimal design approach to select feedback gains and time delay for an active TVA. Since the optimal solution comes in the form of a set, the Pareto set, it comprises all possible performance compromises within the design parameter space. The Pareto optimal solutions provide the robustness against uncertainties in the system. Two examples of delayed feedback controls for the active TVA are presented to demonstrate the effectiveness of the MOP design.

Acknowledgments This work was supported by the Natural Science Foundation of China through the Grants (11172197, 11372271 and 11332008), by the Natural Science Foundation of Zhejiang Province through the Grant (No. LY12A02004) and by the Natural Science Foundation of Tianjin through a key-project grant. L.X.C.'s visit of the University of California, Merced is funded by the China Scholarship Council. The authors would also like to thank the reviewers for their constructive comments that have made this paper much better. References [1] D.J. Inman, Engineering Vibration, Prentice-Hall, Englewood Cliffs, NJ, 1994. [2] J.Q. Sun, M.A. Norris, M.R. Jolly, Passive, adaptive and active tuned vibration absorbers – a survey, The 50th Anniversary Issue of ASME Journal of Vibration and Acoustics and Journal of Mechanical Design 117 (B) (1995) 234–242. [3] D. Filipović, D. Schröder, Bandpass vibration absorber, Journal of Sound and Vibration 214 (3) (1998) 553–566. [4] N. Jalili, L.V. Knowles, W. David, Structural vibration control using an active resonator absorber: modeling and control implementation, Smart Materials and Structures 13 (5) (2004) 998. [5] K. Seto, Y. Furuishi, A study on active dynamic absorber, Proceedings of the 1991 ASME Design Technical Conferences and the 13th Biennial Conference on Mechanical Vibration and Noise, Vol. 38, Miami, FL, 1991, p. 263. [6] N. Olgac, N. Jalili, Modal analysis of flexible beams with delayed resonator vibration absorber: theory and experiments, Journal of Sound and Vibration 218 (2) (1998) 307–331. [7] N. Olgac, D.M. McFarland, B. Holm-Hansen, Position feedback-induced resonance: the delayed resonator, ASME Dynamic System and Control Division Publication 38 (1992) 113–119. [8] N. Olgac, B.T. Holm-Hansen, A novel active vibration absorption technique: delayed resonator, Journal of Sound and Vibration 176 (1) (1994) 93–104. [9] N. Olgac, B. Holmhansen, Design considerations for delayed-resonator vibration absorbers, Journal of Engineering Mechanics 121 (1) (1995) 80–89. [10] N. Olgac, M. Hosek, Active vibration absorption using delayed resonator with relative position measurement, Journal of Vibration and Acoustics 119 (1) (1997) 131–136. [11] N. Olgac, H. Elmali, M. Hosek, M. Renzulli, Active vibration control of distributed systems using delayed resonator with acceleration feedback, Journal of Dynamic Systems, Measurement, and Control 119 (3) (1997) 380–389. [12] N. Olgac, H. Elmali, Analysis and design of delayed resonator in discrete domain, Journal of Vibration and Control 6 (2) (2000) 273–289.

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