Abstract— In this paper, a non-linear control strategy is proposed for handling stroke limits of an active mass driver system. The control force is given by the sum of a linear function of the state of the system and a non-linear dissipative force that becomes effective in the proximity of the physical limit of the actuator. The proposed approach is a generalization of the classic skyhook strategy, where the gain coefficient is modulated by a smooth non-linear function of the displacement of the movable mass relative to structure. The effectiveness of the control law is demonstrated by both experimental tests and numerical simulations, considering a scaled-down five-story frame structure equipped with an active mass driver. The proposed non linear control law proved to be effective in reducing the structural response and avoiding the stroke limit crossing, at the expense of a small increase of structural displacements and a slight increase of structural accelerations.
key words: active structural control, non-linear control strategy, stroke limits, active mass driver, control performance.
I. INTRODUCTION
Active control systems can be highly effective in mitigating the structural response but their use in practical applications is still limited by some technical issues, among which the physical limits of the devices. In the case of strong earthquakes, the limits of the actuators may be exceeded, thus worsening the performance of the controlled structure.
The problem of exceeding the stroke limits is perhaps the most important constraint for application of active mass driver (AMD) systems to actual structures. While the issue of force saturation has been deeply studied in the literature [1-10], the problem of the stroke limits needs a deeper investigation. Nagashima and Shinozaki [11] developed a variable gain feedback (VGF) control for buildings equipped with AMD systems. They proposed a variable feedback gain depending on the trade-off between the reduction of the building response and the amplitude of the mass stroke, that was on-line controlled to keep the stroke within its limits. Yamamoto and Sone [12] utilized a linearly variable gain proportional to an index representing the activity of an AMD, defined as a function of its stroke displacement, the first modal frequency of the controlled building, and the stroke limit of the AMD. Yoshida and Matsumoto [13] proposed a control law for an AMD that took into account the amplitude constraint of the auxiliary mass and was constructed based on a linear saturation control. Within the framework of the State-Dependent Riccati equation, Friedland [14] proposed a strategy suitable to consider the physical constraint in the controller design and in the online computation of the optimal gain matrix.
The Authors have recently started a research program on innovative solutions for structural control [15-18]. In this paper a non-linear control strategy is proposed that is capable to prevent crossing of the stroke limits of an AMD system. To this aim, a skyhook control algorithm is modified by adding to the control force a nonlinear term that increases in the proximity of the stroke bounds. To reduce the impulsive effect of this additional braking term, the variable gain coefficient smoothly varies within the fixed boundaries. The validity of the new approach, that offers the main advantage of being relatively simple to implement, is demonstrated by both numerical simulations and experimental tests.
II. CONTROL SYSTEM DYNAMICS A. Equations of motion
The considered structural system is a n-story planar frame structure equipped with an AMD placed on its top (Fig. 1). Horizontal displacements of the stories relative to the ground are denoted asq ti( ), i = 1, ..., n, while storey masses and stiffness coefficients are denoted as mi and ki, i = 1, ..., n, respectively.
The AMD system is composed of a ball screw that converts the rotational motion of an electric torsional servomotor into the translational motion of a mass ma. The relationship between the displacement of the movable mass relative to the ground
( ) a
q t , the top floor displacement q tn( ) and the rotation of the ball screw ( ) t is:
I. Venanzi is with the Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti 93, 06125, Perugia, Italy (corresponding author: +390755853908; fax: +390755853897; e-mail: [email protected] ).
F. Ubertini L. Ierimontiand A. L. Materazzi are with the Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti 93, 06125, Perugia, Italy (e-mails: [email protected], [email protected], [email protected] ).
Non-linear control strategy for handling stroke limits of an active
mass driver system
Figure 1. Multistory frame structure with AMD.
is the n-dimensional collocation vector of the AMD.
The free response of the system is governed by the following equation of motion, obtained by using Lagrange’s equations [14]:
where M, C and K are nn mass, damping and stiffness matrices of the structure, J is the torsional mass moment of inertia of the ball screw and uT
( )t is the torque provided by the servomotor. This last is regulated by the motor power drive as a function of the setpoint angular velocity. The inertial force, u(t), acting on the movable mass is given by:( ) a a( )
u t m q t (3)
Differentiating (1) with respect to time and using time integration of (3), under the assumption of zero initial conditions, yields the following equation:
which provides the angular velocity to be commanded to the motor for providing the desired control force. A. The proposed non-linear control strategy
consequent exchange of an impulsive force between the mass and the substructure. This can reduce the performance of the control system and produce damages to the control device as well as to the structural system.
A general state controller regulates the feedback control force, u t0( ), as follows:
0( ) ,
u t f y t (5)
where y(t)
q t q t( ), ( )
is the state vector of the system and f y t
, is an appropriate function. This last is, usually, a linear function that can be designed in such a way to guarantee the asymptotic stability of the system and to satisfy some optimal performance criteria. In order to handle the physical limitations imposed on the stroke extension, being impossible to act on xmax, it is necessary to gradually stop the actuator when ( )x t approaches xmax. The easiest way to do it is to introduce in (2) a dissipative force that becomes effective close to the physical limit and is nil elsewhere. Such a force can be obtained with a derivative term with a variable gain GNL moduled by a nonlinear x-dependent coefficient NL( , )x t comprised between 0 and 1:
0( ) ( ) NL NL , ( )
u t u t G x t x t (6)
Equation (6) is in the following referred to as "NLC algorithm" where NLC stands for "Non-Linear Control".
A simple choice forNL( , )x t is the Heaviside step function that assumes a nil value when ( )x t is less than a fraction k (with
0 ≤ k ≤ 1) of the stroke limit xmax and a unit value elsewhere. However, this function is discontinuous for x t( ) kxmax which
determines the application of an abrupt brake force requiring a very large electrical power. In order to solve this problem it is necessary to consider a function which smoothly varies between 0 and 1. In particular NL( , )x t is 1 for x t( ) k x1 max, is 0 for
0 max ( )
x t k x and for intermediate values is given by the following equation:
20 max
1 max 0 max 1
, 1 exp 1
( ) 1 NL x t
x t k x
k x k x
(7)
It is worth noting that NL
x t, given by (7) is an infinitely differentiable function. In particular it does not exhibit any discontinuity and its slope is always finite. This ensures the gradual application of the brake force and the reduction of the associated power. A graphical representation of function NL
x t, is represented in Fig. 2.It can be shown that NL
x t, tends in norm in the L2 functional space to the Heaviside function when k0 and k1 tend to the same value k. It should be noticed that the linearization of (6) in the origin of the state space, here assumed as a fixed equilibrium point of the system, coincides with the linearization of (5). Therefore, the asymptotic stability of the closed-loop system with control algorithm given by (6), also ensures the local asymptotic stability of the NLC algorithm with NL
x t, given by (7). A classic skyhook control algorithm is chosen in this work for specializing f(x) in (5).In the proposed control strategy, the linear term corresponds to a force acting on the movable mass that is proportional to the velocity of the top floor relative to the ground. By regulating u t0( )through this so-called "skyhook control algorithm", (6) is specialized as follows:
( ) L n( ) NL NL , ( )u t G q t G x t x t (8)
where GL is the constant gain of the skyhook strategy.
The demand value of the angular velocity corresponding to (8) is obtained by substituting such equation into (4) and by time integration under the assumption of zero initial conditions. The following control algorithm is thus obtained where integration by parts of the nonlinear term has been carried out:
0
double and single integrations of acceleration signals, respectively.III. EXPERIMENTAL SETUP
The proposed control strategy is tested using a physical model of the structure-AMD system that was designed and constructed in the Laboratory of Structural Dynamics of the Department of Civil and Environmental Engineering of University of Perugia. The experimental set-up shown in Fig. 3 is made of the following components: (i) frame structure; (ii) active mass driver; (iii) monitoring sensors; (iv) controller.
Figure 3. Experimental set-up: a) overview of the frame system, b) detailed view of the AMD, c) joint between motor and ball-screw, d) accelerometer.
a) b)
The test structure is a five-story single-bay S 235 steel frame, shown in Fig. 3 (a), having a total height of about 1.9 m. Each floor is made of 70 x 30 x 2 cm steel plates of about 32.8 kg weight. The structural configuration consists of the use of 0.5 x 3 x 34 cm columns for the lowest two stories and 0.4 x 3 x 34 cm in the remaining ones. The structure is fixed to a support table. The AMD, installed on the top floor, is shown in Fig. 3 (b). It is composed of a 60 cm long ball screw that converts the rotational motion of a Kollmorgen AKM33H AC servomotor into the translational motion of a 4 kg mass block (the 2.5% of the structural mass). The joint between motor and ball screw is shown in Fig. 3 (c). The maximum stroke of the small mass is ± 30 cm, while the ball screw has a diameter of 25 mm and a pitch of 25 mm. The whole AMD system, except for its control unit (drive of the servomotor), is mounted on an aluminum plate designed for the purpose, that also serves as top floor of the structure.
The servomotor is commanded by assigning an angular velocity to the servo-drive of the motor which amplifies such a signal and provides proportional electric current to the motor. An encoder reports the actual status (actual value of the angular velocity) of the motor to the servo drive, which corrects input current by calculating the deviation between commanded and actual status. Because the device actuation is imperfect, the demand value of the angular velocity and its actual value are in general different. However, if the movable mass and the inertia of the screw are small compared to the torque capacity of the servomotor and if the command signal is not excessively fast, imperfect actuation of the device can be neglected and demand and actual values of the angular velocity can be considered approximately coincident.
The position of the mass along the ball screw is measured by means of a JX-P420 linear position transducer. Two optical sensors connected to a logical circuit disable the drive of the motor when the mass exceeds the stroke limits in order to prevent damages of the actuator. Six PCB 393C accelerometers, Fig. 3 (d), are mounted on each floor for the purpose of monitoring structural acceleration.
IV. EXPERIMENTAL TESTS
To investigate the effectiveness of the control strategy, several free vibration tests were carried out. The parameters of the control law were varied in order to examine their effect on the control effectiveness. Experimental results obtained with the proposed non-linear control strategy were compared to those obtained with the classical skyhook algorithm, thus highlighting the effectiveness of the proposed strategy in handling stroke limitations.
In the experiments, the AMD was initially used as an harmonic shaker located at the top floor of the structure and then, after 10 forcing cycles, the AMD was switched to control. The frequency of excitation was 1.8 Hz, close to the first natural frequency of the frame structure. Two different control strategies were compared: 1) the linear “skyhook” control law (LC); 2) the proposed non-linear control law (NLC). As previously described, the corresponding control forces are expressed as:
( ) L n(t) (LC)
u t G q (10)
( ) L n( ) NL NL , ( ) (NLC)
u t G q t G x t x t (11)
where the symbols are described in Section II.
The demand value of the angular velocity for the NLC was expressed by (9). The demand value of the angular velocity for the LC was obtained by posing GNL equal to 0.
In Table I are summarized the parameters of the control algorithm. The gains GL and GNL were set equal to 150 and the stroke limit was assumed to be the 90% of the maximum stroke of the movable mass obtained with the LC strategy. The coefficient k1 was selected as 0.8 and k0 was varied between 0.2 and 0.7.
Fig. 4 shows the top floor displacements of the uncontrolled and controlled system with LC and NLC strategies. It can be observed that during the loading phase the structural displacements increase while, when the control is switched on, the structural response is rapidly decreased by both the adopted control approaches. Fig. 5 shows the results in terms of AMD displacements (a) and top floor displacements (b) obtained with the LC and the NLC algorithms for k0 = 0.2 and k1 = 0.8. It is possible to observe that, by adopting the NLC strategy, the AMD stroke never exceeds xmax. Moreover, the classical control algorithm is not self re-centering while the non linear algorithm contributes to limiting the residual AMD displacement, thus avoiding the need for manual or automatic re-centering after the seismic event.
(a) (b) TABLE I. PARAMETERS OF THE CONTROL ALGORITHM
Parameter Value
GL 150
GNL 150
xmax 6.8 cm
k1 0.8
k0 0.2-0.7
Figure 4. Top floor displacements for the uncontrolled (UNC) and controlled systems with LC and NLC ( k0 = 0.2) strategies.
Fig. 6 (a) shows the non linear term of the velocity demand value that is activated when the AMD stroke exceeds the value k0xmax. The non linear term can be obtained from (10) as follows:
, ( ) 2( ) NL NL NL
a G x t x t t
l m
(12)
Fig. 6 (b) shows the velocity demand value obtained with both LC and NLC strategies. It is visible a peak of the AMD velocity in correspondence of the first braking of the movable mass. After the first peak, the velocity demand value required by the two algorithms is almost comparable.
a) b)
Figure 6. Non-linear velocity demand for the NLC system (a); comparison between the velocity demand values obtained
with the LC and NLC systems (b).
The control effectiveness of the proposed NLC strategy was also investigated by means of a series of performance indices. This allowed comparing the classical control law with the non-linear proposed strategy:
0
where t0 is the time when the AMD is switched to control; J1 is the ratio of the maximum relative displacements between the AMD qa(t) and the top floor qn(t) obtained with the NLC and the LC strategies; J2 is the ratio of the maximum floor displacements qn(t) obtained with the NLC and the LC strategies; J3 is the ratio of the maximum floor accelerations q ti( ) obtained with the NLC and the LC strategies; J4 is the ratio of the maximum velocity demand value D( )t obtained with the NLC and the LC strategies.
In Table II are shown the performance indices corresponding to experimental responses obtained for different values of parameter k0 . The value k0 = 0.8 was not considered because, as observed in Section 3, when k0 = k1, NL
x t, coincides withHeaviside’s step function which cannot be reproduced in practice.
TABLE II. PERFORMANCE INDICES COMPUTED ADOPTING THE EXPERIMANTAL RESULTS
k0 J1 J2 J3 J
0.2 0.749 0.987 0.995 4.642
0.3 0.767 1.001 1.004 4.803
0.4 0.843 1.012 1.208 5.117
0.5 0.857 1.008 1.268 5.332
0.6 0.862 1.005 1.214 5.541
0.7 0.891 1.059 1.226 5.603
V. NUMERICAL SIMULATIONS A. Numerical modeling
In order to better interpret the experimental results and further discuss the effectiveness of the control strategy several numerical tests were performed in free and forced vibration conditions. Preliminarily, a numerical model was built and tuned to the free vibration response of the uncontrolled system.
A modal identification allowed to obtain an accurate numerical model of the system, considered as a shear-type structure having 5 degrees of freedom. The signals provided by 5 accelerometers located on the floors, recorded with a sampling frequency of 1 kHz and a total duration of 1 hour, were used for modal identification. The structural response signals were acquired with excitation provided by microtremors in the laboratory.
The Frequency Domain Decomposition (FDD) was used to perform the modal identification after decimation of the measured time history signals to 50 Hz and using a frequency resolution to compute the power spectral density matrix of the measurements equal to 0.0122 Hz. The results of modal identification are summarized in Table III. In the same Table are also reported the modal damping ratios obtained by analyzing the envelopes of the free decay responses of the structural modes. In Fig. 7 is shown the free vibration response of the uncontrolled structure and the estimated decay envelopes for the first two modes. Single mode responses are obtained by harmonically exciting the structure with the AMD working as a shaker providing excitation in resonance with a specific mode. As the modal damping ratios vary slightly with the amplitude of vibration, their values are estimated in intermediate ranges of the response.
After identifying the dynamic characteristics of the physical system, a model tuning was performed to obtain the masses and stiffness coefficients of the frame to be adopted in the numerical model, as detailed in [15]. In Table IV are reported the
updated floors’ stiffness and mass coefficients.
TABLE III. IDENTIFIED MODAL CHARACTERISTICS OF THE SYSTEM
Mode Natural frequency
(Hz)
Modal damping ratio (%)
1 1.76 0.3
2 4.92 0.4
3 7.34 1.1
4 9.62 1.2
5 12.15 1.3
TABLE IV. PARAMETERS OF THE UPDATED NUMERICAL MODEL
Story kopt
(N/m)
mopt
(kg)
1 17286 31.93
2 16404 32.19
3 8619 31.92
4 8698 32.33
Figure 7. Free vibration responses of the uncontrolled structure and estimated decay envelopes for the 1st mode (a) and the second mode (b).
After identifying the dynamic characteristics of the physical system, a model tuning was performed to obtain the masses and stiffness coefficients of the frame to be adopted in the numerical model, as detailed in [15]. In Table IV are reported the
updated floors’ stiffness and mass coefficients.
B. Free vibration tests
The experimental tests were reproduced numerically applying at the top of the structure a sinusoidal inertial force corresponding to the acceleration of the AMD.
Fig. 8 shows the comparison between numerical and experimental AMD and top floor displacements for both LC (GL = 150) and NLC strategies (GNL = 150, k0 = 0.2, k1 = 0.8). The comparison is limited to the controlled part of the response. The experimental and numerical results are very close to each other. The smaller AMD displacement obtained with the NLC is due to the fact that the numerical model does not consider the imperfect actuation of the physical system. Also the comparison between the numerical and experimental top floor displacements, shown in Fig. 8, shows a good correspondence. Fig. 9 reports the comparison between the numerically predicted control force and top floor displacements obtained with LC and NLC strategies. The NLC algorithm leads to an important increase in the control force at the beginning of the actuation. In Table V are summarized the performance indices, introduced in Section IV, computed by adopting the numerical results. The index J5 is computed by using the control force instead of the velocity demand value, as follows:
0
0
5
max ( ) max ( )
NLC
t t
LC
t t u t J
u t
(17)
where ( )u t is given by (12-13).
Figure 8. Numerical and experimental AMD displacements for LC (a) and NLC (b); numerical and experimental top floor displacements for LC (c) and NLC (d).
TABLE V. PERFORMANCE INDICES COMPUTED ADOPTING THE NUMERICAL RESULTS,FREE VIBRATION TEST
k0 J1 J2 J3 J5
0.2 0.585 1.000 1.000 1.943
0.3 0.619 1.000 1.000 2.412
0.4 0.651 1.000 1.000 2.392
0.5 0.681 1.000 1.000 2.737
0.6 0.711 1.000 1.000 3.268
a) b) a) b)
Figure 9. Comparison between the LC and NLC strategies in terms of control force (a) and top floor displacement (b).
C. Forced vibration tests
Forced vibration tests consisted in the application of a sinusoidal motion at the base of the structure. The frequency of the harmonic input was increased from 1 to 2.5 Hz with step increments of 0.05 Hz, thus covering the region of the resonance with the first mode.
Tests were carried out for both uncontrolled and controlled structures, comparing LC and NLC strategies. The parameters of the control algorithms are the same as in Table I, except for the stroke limit xmax that was set equal to 0.04 m according to the different maximum stroke of the movable mass obtained with the LC strategy.
Fig. 10 shows the results of the numerical simulations in terms of AMD displacements (a) and top floor displacements (b) at an excitation frequency of 1.8 Hz. These results confirm the effectiveness of the proposed non linear control law in keeping the AMD stroke below its limit. On the other hand, an aggravation in terms of structural response is accepted, in order to ensure proper device operation.
Figs. 11-12-13 show the results in terms of Frequency Response Functions (FRF) for top displacements q tn( ), AMD displacements ( )x t , and control force ( )u t .
The Frequency Response Functions of the selected variables are defined as follows:
where j is the analyzed frequency;q tn( )is the top floor displacement evaluated for uncontrolled, controlled LC and controlled NLC cases; ( )x t is the AMD displacement for NLC and LC cases; ( )u t is the control force, computed with both controlled strategies;qUNCn ( )t represents top floor displacements for uncontrolled system;xLC( )t is the AMD displacement for controlled system with LC;
u
NLC( )
t
is the control force value for controlled system with NLC.
Figure 12. Frequency response functions of the AMD displacements for LC and NLC strategies.
Figure 13. Frequency response functions of the control force for LC and NLC strategies.
The results presented in Fig. 11 confirm that the control system is highly effective in reducing the structural response in the
region of the first modal resonance for both controlled strategies. Fig. 12 confirms that the NLC strategy handles the stroke of
the mass keeping it below the values obtained from LC algorithm in the whole investigated frequency. According to what observed before, the numerical results shown in Fig. 13 highlight that the maximum control power demand is higher in the case of the NLC strategy than in the case of the LC strategy.
In Table VI are reported the performance indices, defined in Section IV and V, that compare NLC and LC strategies. The results in Table VI confirm that also in forced vibration tests, the effectiveness of the proposed control law in reducing the AMD stroke is reduced as k0 increases. Consequently, an increase of the control force is observed (as J5 increases) and indices J2 and J3 are greater than 1 for all analyzed cases, highlighting that the NLC algorithm determines a reduction in control effectiveness, differently from what observed in free vibration.
Figs. 14-15-16 show the differences between performance indices J1, J2 and J5 evaluated numerically in free and forced vibration tests, as a function of k0. These results highlight that the percentage differences between the indices computed in free and forced vibration decrease with the increase of k0.
Figure 15. Index J2 in comparison between free and forced vibrations tests (k0 = 0.2-0.7).
Figure 16. Index J5 in comparison between free and forced vibrations tests (k0 = 0.2-0.7).
TABLE VI. PERFORMANCE INDICES COMPUTED ADOPTING THE NUMERICAL RESULTS,FORCED VIBRATIONS TESTS
k0 J1 J2 J3 J5
0.2 0.737 1.39 1.93 3.14
0.3 0.739 1.35 1.89 3.29
0.4 0.743 1.30 1.87 3.45
0.5 0.748 1.26 1.84 3.61
0.6 0.754 1.22 1.78 3.73
VI. CONCLUSIONS
In this paper, a non-linear control strategy based on a modified direct velocity feedback algorithm is proposed for handling stroke limits of an AMD system. In particular, a non-linear braking term proportional to the relative AMD velocity is included in the control algorithm in order to slowdown the device in the proximity of the stroke limits.
Experimental tests were carried out on a scaled-down five-story building equipped with an AMD under free vibrations. Several numerical tests were also been performed in free and forced vibrations conditions.
The proposed non linear control law proved to be effective in reducing the structural response and avoiding the stroke limit crossing, at the expense of a small increase of structural displacements and a slight increase of structural accelerations. A parametric analysis showed that an increase of the ratio (k1/k0) between the parameters defining the nonlinear restoring force results in an improved performance in terms of braking efficacy. On the contrary, the same braking efficacy can be obtained with a lower k1/k0 ratio and a higher gain GNL, at the expense of an increase in the structural response, especially in terms of accelerations. The optimal choice of the k1/k0 ratio should be obtained as a compromise between the need of limiting the power demand and that of limiting the structural response. In free vibration conditions, both numerical and experimental tests showed that the growth in power demand with respect to the linear case is important only for the first braking peak. The results obtained in forced vibration conditions showed a deterioration of the structural response and an increased power demand with respect to the free vibration response.
ACKNOWLEDGMENT
The Authors gratefully acknowledge the financial support of the ”Cassa di Risparmio di Perugia” Foundation that funded this study through the project ”Development of active control systems for the response mitigation under seismic excitation” (Project Code 2010.011.0490).
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