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Active vibration damping based on neural network theory
Materials Science and Engineering A 442 (2006) 547–550
Active vibration damping based on neural network theory Hisashi Kawabe a,∗ , Naoki Tsukiyama b , Kazunobu Yoshida c a
Faculty of Engineering, Hiroshima Institute of Technology, 2-1-1 miyake, Saekiku, Hiroshima 731-5143, Japan b Graduate School, Hiroshima Institute of Technology, 2-1-1 miyake, Saekiku, Hiroshima 731-5143, Japan c Interdisciplinary Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu, Matsue 690-8504, Japan Received 5 August 2005; received in revised form 10 January 2006; accepted 3 February 2006
Abstract The possibility of utilizing neural networks (NN) theory for active vibration control in a longitudinal cantilevered-beam system is investigated by simulation and experiment. The feedback control system providing active vibration damping is constructed with a three-layer-type neural network controller, a strain gauge sensor, and an actuator generating electro-magnetically interactive control force. It is found that the active damping effect using the NN controller is obtained as Q−1 max = 0.144 by the specific damping capacity criterion, which corresponds to about 6.2 times the maximum damping effect observed in the Fe-based ferromagnetic high damping metal called SIA, and that the NN vibration control system is also robust against some parameter variation. © 2006 Elsevier B.V. All rights reserved. Keywords: Active vibration damping; Neural networks; High damping metal; Vibration control; Feedback controller; Specific damping capacity
1. Introduction So-called high damping alloys [1] were developed, and successful approaches to the anti-vibration by the material damping effect were introduced accordingly [2]. However, such passive vibration damping techniques [3] are not well adopted to practical machine structure elements as expected, partly because of a practical limitation in the possibility of obtaining higher damping capacity due to the inherent conditions concerned with the construction materials. On the other hand, with the development of control theory, increasing attention has been focused on active vibration damping techniques, which are basically constructed with sensors, actuators, and a feedback controller to compensate damping and modulus properties, with successful applications [4]. Active damping systems have the advantages that they can supply, by feedback, the damping energy when required as well as dissipate it, and that the mechanical properties (damping and modulus) can be arbitrarily designed in a range depending on the control law adopted. In this paper, we propose an active damping technique using a neural network (NN) controller, and investigate the effect of the
∗
Corresponding author. Tel.: +81 82 921 4539; fax: +81 82 923 9296. E-mail address: [email protected] (H. Kawabe).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.02.234
NN damping in comparison with the material damping obtained in the ferromagnetic high damping metal SIA (an abbreviation of Toshiba’s commercial damping metal Silentalloy [2]). 2. Modeling and control law Fig. 1 shows a schematic illustration of an active damping control system consisting of a longitudinal cantileveredbeam. The vibration displacement x1 (t) is detected by the strain gauge attached near the fixed end, and is fed back to the controller through an A/D converter, and in turn the controller supplies the vibration control signal u(t) through a D/A converter. While the controlled object is a mechanically flexible system having multi-elastic vibration modes, a single-degree-offreedom lumped parameter model is adopted to simplify the real system. 2.1. Modeling of controlled object As shown in Fig. 1, the controlled object is approximated as a lumped parameter system, where m (kg), k (N/m), and c (N s/m) are the mass, the spring constant, and the viscous damping coefficient, respectively. Let x1 be the mass displacement, and define the state variT able as x = [ x1 x2 (= x˙ 1 ) ] . Then the state equation of the
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H. Kawabe et al. / Materials Science and Engineering A 442 (2006) 547–550
acterized with the numerical connection weights between the units and the inputs to the networks. The weighting functions wji and vkj are altered by a learning method that adjusts their values using the back propagation method based on the steepest descent algorithms given by ⎫ J ⎪ ⎪ vkj (k + 1) = −α vkj (k) ⎬ (3) J ⎪ ⎪ ⎭ wji (k + 1) = −α wji (k) where the differences at the k-th sampling time are written by ⎫ J = J(k) − J(k − 1) ⎪ ⎬ vkj (k) = vkj (k) − vkj (k − 1) (4) ⎪ ⎭ wji (k) = wji (k) − wji (k − 1) Here, the performance index J and the error e are defined as 1 eTe, . 2 e = xˆ m − x J=
(5)
The learning control is after all given by uL (k) = Kn · Ok (k)
Fig. 1. Schematic illustration for the overall experimental system.
controlled object is represented by x˙ = Ax + bu , y = Cx
(6)
where (1)
where the matrices are as follows: 0 1 0 A= , b= , C = [ 1 0 ]. (2) −k/m −c/m kf /m
⎫ vkj (k) · Hj (k), ⎪ ⎪ ⎪ ⎪ ⎪ j ⎬ Hj (k) = wji (k) · Ii (k), ⎪ ⎪ ⎪ i ⎪ ⎪ ⎭ Ii (k) = ei (k) · Ni Ok (k) =
(7)
and where Ni is a compressive coefficient.
The interactive applied force (control input) u is produced by the feedback control law, u = kf x, where kf is the feedback gain to be designed. 2.2. Construction of a neural network control system The learning control uL , adopted here, is given by a threelayer neural network as shown in Fig. 2. The structure is char-
Fig. 2. Structure of three-layer neural network.
Fig. 3. Active vibration control system (including a full state observer) based on a neural network controller and a model-based one.
H. Kawabe et al. / Materials Science and Engineering A 442 (2006) 547–550
549
Table 1 Initial values of the weighting factors wji , vkj , and necessary factors Ni , Kn , and α wji vkj Factors
w11 = 0.5, w12 = −0.5, w21 = 0.1, w22 = −0.3 w31 = 0.7, w32 = −0.1, w41 = −0.6, w42 = 0.4 w51 = 0.8, w52 = 0.1, w61 = −0.1, w62 = 0.3 v11 = 0.2, v12 = −0.4, v13 = 0.9 v14 = −0.8, v15 = −0.3, v16 = −0.5 N1 = 0.00065, N2 = 0.00065, Kn = 1.0, α = 0.20
2.3. Design of a full state observer Only the displacement x1 (t) can be measured, so the other variable x2 = x˙ 1 needs to be estimated by constructing an observer. A full state observer [5] estimating the state x(t) is employed, which has the form: ˆx+B ˆ uˆ xˆ˙ = Aˆ , (8) ˆ xˆ + D ˆ uˆ yˆ = C
Fig. 4. Vibration control effect by the NN controller: (a) without parameter variation and (b) with variation m.
ˆ 2×2 = A − LC, B ˆ 2×2 = [I2×2 ], ˆ 2×1 = [ L B ], C where A ˆ D2×1 = [02×1 ]. The overall control system including the full state observer is shown in Fig. 3. 3. Experimental method As shown in Fig. 1, a longitudinal type cantilever beam (300 mm × 35 mm × 3 mm and made of steel) is adopted as the controlled object. The actuating control u(t) is generated as an electro-magnetic force between the lumped mass m1 (=1.75 kg) and the auxiliary mass m2 (=0.15 kg) at its free end. The observer gain L = [37.36 35.64] is used so that the complex poles of observer are placed at λ1,2 = −18.74 ± i8.95. Initial values of the weighting factors wji , vkj and the necessary factors Ni , Kn , and α for computing the NN system are shown in Table 1.
Fig. 5. Amplitude-dependent active damping effect using the NN controller.
4. Results and discussion Fig. 4 shows the controlled vibration waveform in the real vibration control system using the neural network controller. Fig. 4(a) shows the control effect for the system without parameter variations, while Fig. 4(b) shows the vibration control effect for the system with weight parameter variation m = 0.5 kg. By making a comparison of Fig. 4(a) with (b), it is found that
Fig. 6. Robust NN Control effect against applied impulsive disturbances.
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H. Kawabe et al. / Materials Science and Engineering A 442 (2006) 547–550
the vibration-suppression effect by the active damping is more noticeable irrespective of the presence of the parameter variation. As shown in Fig. 5, the vibration control effects can be represented as a function of vibration amplitude, using the specific damping capacity criterion [2]
2 2 − A A 1 n n+1 Q−1 = , (9) 2π A2n where An and An+1 are the n-th and (n + 1)-th amplitudes of the decaying waveform. From a careful inspection, there can be seen a tendency that the damping capacity becomes a slightly lower for smaller amplitudes (Q−1 ≈ 0.144–0.135 for smaller amplitude), but good active damping characteristics are obtained over the whole amplitudes. For reference, the active vibration damping effect based on an LQ controller [6] is also investigated with the result that a fairly well controlled vibration damping can be obtained, which is similar to the above mentioned NN system. However, the degradation of the control performance due to the parameter variation is a little more noticeable as compared with the NN system. The active damping effect via the NN controller corresponds to about 6.2 times the maximum damping effect observed in the ferromagnetic type high damping metal SIA. Fig. 6 shows the robust NN Control effect against applied impulsive disturbances. A good stable property can be maintained against some disturbances.
5. Conclusions The active vibration damping characteristics have been investigated in a cantilever beam vibration system using a feedback controller based on the neural network learning theory. (1) Fairly satisfactory active damping effect is obtained as Q−1 max ≈ 0.144 by the specific damping damping capacity criterion. (2) The NN-active damping is found to be robust against some parameter variation and some impulsive disturbances. (3) The NN vibration damping corresponds to about 6.2 times the maximum damping effect observed in the Fe-based ferromagnetic high damping metal called SIA. References [1] B.J. Lazan, Damping of Materials and Members in Structural Mechanics, Pergamon Press, New York, 1968, pp. 181–182. [2] H. Kawabe, K. Kuwahara, Trans. JIM 22-5 (1981) 301–308. [3] T. Hidaka, S. Oda, H. Kawabe, U. Sogabe, K. Yoshida, Mechanical Dynamics, Asakura, Japan, 2000, pp. 123–124 (in Japanese). [4] K. Seto, K. Suzuki, J. Jpn. Soc. Mech. Eng. 89–811 (1986) 635–642. [5] K. Furuta, S. Kawaji, T. Mita, S. Hara, Control of Mechanical System, O-mu, Japan, 1984, pp. 67–68 (in Japanese). [6] K. Okuyama, H. Kawabe, K. Yoshida, Y. Yukio, Takemor S. Noritsugi, Control Engineering, Asakura, Japan, 2001, pp. 88–89 (in Japanese).
Active vibration damping based on neural network theory Hisashi Kawabe a,∗ , Naoki Tsukiyama b , Kazunobu Yoshida c a
Faculty of Engineering, Hiroshima Institute of Technology, 2-1-1 miyake, Saekiku, Hiroshima 731-5143, Japan b Graduate School, Hiroshima Institute of Technology, 2-1-1 miyake, Saekiku, Hiroshima 731-5143, Japan c Interdisciplinary Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu, Matsue 690-8504, Japan Received 5 August 2005; received in revised form 10 January 2006; accepted 3 February 2006
Abstract The possibility of utilizing neural networks (NN) theory for active vibration control in a longitudinal cantilevered-beam system is investigated by simulation and experiment. The feedback control system providing active vibration damping is constructed with a three-layer-type neural network controller, a strain gauge sensor, and an actuator generating electro-magnetically interactive control force. It is found that the active damping effect using the NN controller is obtained as Q−1 max = 0.144 by the specific damping capacity criterion, which corresponds to about 6.2 times the maximum damping effect observed in the Fe-based ferromagnetic high damping metal called SIA, and that the NN vibration control system is also robust against some parameter variation. © 2006 Elsevier B.V. All rights reserved. Keywords: Active vibration damping; Neural networks; High damping metal; Vibration control; Feedback controller; Specific damping capacity
1. Introduction So-called high damping alloys [1] were developed, and successful approaches to the anti-vibration by the material damping effect were introduced accordingly [2]. However, such passive vibration damping techniques [3] are not well adopted to practical machine structure elements as expected, partly because of a practical limitation in the possibility of obtaining higher damping capacity due to the inherent conditions concerned with the construction materials. On the other hand, with the development of control theory, increasing attention has been focused on active vibration damping techniques, which are basically constructed with sensors, actuators, and a feedback controller to compensate damping and modulus properties, with successful applications [4]. Active damping systems have the advantages that they can supply, by feedback, the damping energy when required as well as dissipate it, and that the mechanical properties (damping and modulus) can be arbitrarily designed in a range depending on the control law adopted. In this paper, we propose an active damping technique using a neural network (NN) controller, and investigate the effect of the
∗
Corresponding author. Tel.: +81 82 921 4539; fax: +81 82 923 9296. E-mail address: [email protected] (H. Kawabe).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.02.234
NN damping in comparison with the material damping obtained in the ferromagnetic high damping metal SIA (an abbreviation of Toshiba’s commercial damping metal Silentalloy [2]). 2. Modeling and control law Fig. 1 shows a schematic illustration of an active damping control system consisting of a longitudinal cantileveredbeam. The vibration displacement x1 (t) is detected by the strain gauge attached near the fixed end, and is fed back to the controller through an A/D converter, and in turn the controller supplies the vibration control signal u(t) through a D/A converter. While the controlled object is a mechanically flexible system having multi-elastic vibration modes, a single-degree-offreedom lumped parameter model is adopted to simplify the real system. 2.1. Modeling of controlled object As shown in Fig. 1, the controlled object is approximated as a lumped parameter system, where m (kg), k (N/m), and c (N s/m) are the mass, the spring constant, and the viscous damping coefficient, respectively. Let x1 be the mass displacement, and define the state variT able as x = [ x1 x2 (= x˙ 1 ) ] . Then the state equation of the
548
H. Kawabe et al. / Materials Science and Engineering A 442 (2006) 547–550
acterized with the numerical connection weights between the units and the inputs to the networks. The weighting functions wji and vkj are altered by a learning method that adjusts their values using the back propagation method based on the steepest descent algorithms given by ⎫ J ⎪ ⎪ vkj (k + 1) = −α vkj (k) ⎬ (3) J ⎪ ⎪ ⎭ wji (k + 1) = −α wji (k) where the differences at the k-th sampling time are written by ⎫ J = J(k) − J(k − 1) ⎪ ⎬ vkj (k) = vkj (k) − vkj (k − 1) (4) ⎪ ⎭ wji (k) = wji (k) − wji (k − 1) Here, the performance index J and the error e are defined as 1 eTe, . 2 e = xˆ m − x J=
(5)
The learning control is after all given by uL (k) = Kn · Ok (k)
Fig. 1. Schematic illustration for the overall experimental system.
controlled object is represented by x˙ = Ax + bu , y = Cx
(6)
where (1)
where the matrices are as follows: 0 1 0 A= , b= , C = [ 1 0 ]. (2) −k/m −c/m kf /m
⎫ vkj (k) · Hj (k), ⎪ ⎪ ⎪ ⎪ ⎪ j ⎬ Hj (k) = wji (k) · Ii (k), ⎪ ⎪ ⎪ i ⎪ ⎪ ⎭ Ii (k) = ei (k) · Ni Ok (k) =
(7)
and where Ni is a compressive coefficient.
The interactive applied force (control input) u is produced by the feedback control law, u = kf x, where kf is the feedback gain to be designed. 2.2. Construction of a neural network control system The learning control uL , adopted here, is given by a threelayer neural network as shown in Fig. 2. The structure is char-
Fig. 2. Structure of three-layer neural network.
Fig. 3. Active vibration control system (including a full state observer) based on a neural network controller and a model-based one.
H. Kawabe et al. / Materials Science and Engineering A 442 (2006) 547–550
549
Table 1 Initial values of the weighting factors wji , vkj , and necessary factors Ni , Kn , and α wji vkj Factors
w11 = 0.5, w12 = −0.5, w21 = 0.1, w22 = −0.3 w31 = 0.7, w32 = −0.1, w41 = −0.6, w42 = 0.4 w51 = 0.8, w52 = 0.1, w61 = −0.1, w62 = 0.3 v11 = 0.2, v12 = −0.4, v13 = 0.9 v14 = −0.8, v15 = −0.3, v16 = −0.5 N1 = 0.00065, N2 = 0.00065, Kn = 1.0, α = 0.20
2.3. Design of a full state observer Only the displacement x1 (t) can be measured, so the other variable x2 = x˙ 1 needs to be estimated by constructing an observer. A full state observer [5] estimating the state x(t) is employed, which has the form: ˆx+B ˆ uˆ xˆ˙ = Aˆ , (8) ˆ xˆ + D ˆ uˆ yˆ = C
Fig. 4. Vibration control effect by the NN controller: (a) without parameter variation and (b) with variation m.
ˆ 2×2 = A − LC, B ˆ 2×2 = [I2×2 ], ˆ 2×1 = [ L B ], C where A ˆ D2×1 = [02×1 ]. The overall control system including the full state observer is shown in Fig. 3. 3. Experimental method As shown in Fig. 1, a longitudinal type cantilever beam (300 mm × 35 mm × 3 mm and made of steel) is adopted as the controlled object. The actuating control u(t) is generated as an electro-magnetic force between the lumped mass m1 (=1.75 kg) and the auxiliary mass m2 (=0.15 kg) at its free end. The observer gain L = [37.36 35.64] is used so that the complex poles of observer are placed at λ1,2 = −18.74 ± i8.95. Initial values of the weighting factors wji , vkj and the necessary factors Ni , Kn , and α for computing the NN system are shown in Table 1.
Fig. 5. Amplitude-dependent active damping effect using the NN controller.
4. Results and discussion Fig. 4 shows the controlled vibration waveform in the real vibration control system using the neural network controller. Fig. 4(a) shows the control effect for the system without parameter variations, while Fig. 4(b) shows the vibration control effect for the system with weight parameter variation m = 0.5 kg. By making a comparison of Fig. 4(a) with (b), it is found that
Fig. 6. Robust NN Control effect against applied impulsive disturbances.
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H. Kawabe et al. / Materials Science and Engineering A 442 (2006) 547–550
the vibration-suppression effect by the active damping is more noticeable irrespective of the presence of the parameter variation. As shown in Fig. 5, the vibration control effects can be represented as a function of vibration amplitude, using the specific damping capacity criterion [2]
2 2 − A A 1 n n+1 Q−1 = , (9) 2π A2n where An and An+1 are the n-th and (n + 1)-th amplitudes of the decaying waveform. From a careful inspection, there can be seen a tendency that the damping capacity becomes a slightly lower for smaller amplitudes (Q−1 ≈ 0.144–0.135 for smaller amplitude), but good active damping characteristics are obtained over the whole amplitudes. For reference, the active vibration damping effect based on an LQ controller [6] is also investigated with the result that a fairly well controlled vibration damping can be obtained, which is similar to the above mentioned NN system. However, the degradation of the control performance due to the parameter variation is a little more noticeable as compared with the NN system. The active damping effect via the NN controller corresponds to about 6.2 times the maximum damping effect observed in the ferromagnetic type high damping metal SIA. Fig. 6 shows the robust NN Control effect against applied impulsive disturbances. A good stable property can be maintained against some disturbances.
5. Conclusions The active vibration damping characteristics have been investigated in a cantilever beam vibration system using a feedback controller based on the neural network learning theory. (1) Fairly satisfactory active damping effect is obtained as Q−1 max ≈ 0.144 by the specific damping damping capacity criterion. (2) The NN-active damping is found to be robust against some parameter variation and some impulsive disturbances. (3) The NN vibration damping corresponds to about 6.2 times the maximum damping effect observed in the Fe-based ferromagnetic high damping metal called SIA. References [1] B.J. Lazan, Damping of Materials and Members in Structural Mechanics, Pergamon Press, New York, 1968, pp. 181–182. [2] H. Kawabe, K. Kuwahara, Trans. JIM 22-5 (1981) 301–308. [3] T. Hidaka, S. Oda, H. Kawabe, U. Sogabe, K. Yoshida, Mechanical Dynamics, Asakura, Japan, 2000, pp. 123–124 (in Japanese). [4] K. Seto, K. Suzuki, J. Jpn. Soc. Mech. Eng. 89–811 (1986) 635–642. [5] K. Furuta, S. Kawaji, T. Mita, S. Hara, Control of Mechanical System, O-mu, Japan, 1984, pp. 67–68 (in Japanese). [6] K. Okuyama, H. Kawabe, K. Yoshida, Y. Yukio, Takemor S. Noritsugi, Control Engineering, Asakura, Japan, 2001, pp. 88–89 (in Japanese).