H_∞ control in the frequency domain for a semi-active floor isolation system

Abstract

A floor isolation system installed in a single floor or room in a fixed base structure is designed to protect equipment. With this configuration, the input motions to the floor isolation from the ground motions are filtered by the structure, leaving the majority of the frequency content of the input motion lower than the predominant frequency of the structure. The floor isolation system should minimize the acceleration to protect equipment; however, displacement must also be limited to save floor space, especially with long period motion. Semi-active control with an H_∞ control was adopted for the floor isolation system and a new input shaping filter was developed to account for the input motion characteristics and enhance the effectiveness of the H_∞ control. A series of shake table tests for a semi-active floor isolation system using rolling pendulum isolators and a magnetic-rheological damper were performed to validate the H_∞ control. Passive control using an oil damper was also tested for comparison. The test results show that the H_∞ control effectively reduced acceleration for short period motions with frequencies close to the predominant frequency of the structure, as well as effectively reduced displacement for long period motions with frequencies close to the natural frequency of the floor isolation system. The H_∞ control algorithm proved to be more advantageous than passive control because of its capacity to adjust control strategies according to the different motion frequency characteristics.

Keywords

semi-active / floor isolation / H_∞ control / MR damper / shaping filter / shaking table test

Introduction

Floor isolation systems designed for a single floor or room of the fixed base structure are a cost effective alternative [1-4] to base isolation systems [5-7] for protecting sensitive equipment. To protect equipment from damage, the acceleration of the isolated floor must be reduced. However, as space for a floor isolation system is often limited, small clearance distances between the floor isolation system and surrounding walls are desirable to maximize the usable space on the isolated floor. These requirements must be taken into account when designing floor isolation.

The floor isolation system may be placed on a relatively higher floor of the structure. In this configuration, it is discovered that the input motion to the floor isolation system, i.e., the response of the floor on which the floor isolation system is installed, has notably different characteristics from the ground motion to the structure. Usually, the input motion is amplified from the ground to floor [7,8] and filtered by the structure. When a short period ground motion with high frequency attacks the structure, it will result in an input motion to the floor isolation having a dominant frequency that is close to the predominant frequency of the structure, and this frequency component will be amplified. The frequencies higher than the predominant frequency of the structure will be significantly filtered out. However, when along period motion with low frequency, here defined as a motion whose dominant frequency is 0.2 to 0.5 Hz (2 to 5 s), attacks the structure, the low frequency component will be transferred through the structure and excites the floor isolation.

As ground motion has high variability, the floor isolation may be subjected to motions with different frequency components. Motions with high frequency tend to cause large acceleration of the floor isolation system because of the high amplitude, while motions with low frequency tend to cause large displacement because of resonance. A passive type control is found to be difficult to optimally control the floor response under motions with different frequency characteristics [9]. To reduce the acceleration so as to protect equipment on the floor isolation, and reduce the displacement to maximize usable space, a semi-active controlled floor isolation system with an H_∞ control is designed to create a system which can deal with the variability of the excitation.

H_∞ control is a frequency domain control method that allows the designer to directly deal with the input motion characteristics and specify the disturbance attenuation over a desired frequency range. Generally, the more information about the controlled system fed to the controller, the more effective the control can be. When little is known about the input motion, it is typically represented as a white noise or a band-limited white noise process [10,11]. However, this simplification would decrease the efficiency of H_∞ control significantly. Due to the unique configuration of floor isolation, the input motion to the floor isolation has unique characteristics. To improve the efficiency of H_∞ control so that it can deal with both short and long period motions, a new input motion shaping filter is developed in this study to consider the frequency characteristics of the input motion to the floor isolation system.

This paper presents the design of an H_∞ control in the frequency domain for a semi-active floor isolation system. A series of shaking table tests were conducted to verify the designed control. The configuration of the semi-active floor isolation system using a magnetic-rheological (MR) damper is presented in Section 2. Section 3 introduces the design of H_∞ control with the shaping filter to consider the input motion characteristics. The shaking table program and the test results are shown in Section 4. A passive floor isolation system was also designed for comparison.

Floor isolation system and modeling

Floor isolation system configuration

Figure 1 shows the schematics for a base isolation system and a floor isolation system. While base isolation systems are designed for an entire building, floor isolation systems can be applied to a single room or one floor of a fixed base structure. The floor isolation considered in this study is located on the top floor of a RC building. It is designed to protect a group of sensitive or expensive equipment from potential earthquake damage. The floor isolation system contains a rolling pendulum system to ensure the flexibility. The designed natural period of the floor isolation system is 3.0 s. A damping system is needed to provide the control force. Depending on the control method, a passive or semi-active type damper can be used.

Input motion characteristics

With the configuration of floor isolation system shown in Fig. 1, the input motion to the floor isolation system is the response of the top floor, which is distinct from the ground motion to the structure. When the ground motion is transferred from the ground to the floor where the floor isolation is installed, the ground motion is physically filtered by the structure.

When the ground motion dominated by a high frequency (short period) attacks the structure, it will result in the input motion to the floor isolation with the dominant frequency close to the predominant frequency of the structure. The input motion can also be amplified several times from the ground motion because of the resonance effect with the structure [3,7].This kind of motion tends to cause large acceleration due to the large magnitude. When the ground motion dominated by a low frequency (long period), here defined as a motion whose dominant frequency is 0.2 to 0.5 Hz (2 to 5 s), attacks the structure, the low frequency component will be transferred through the structure. Although the low frequency is not amplified from the ground, it will still cause large response, especially the displacement, due to the resonance with the floor isolation system.

In summary, the structure significantly filters out frequency components in the ground motion higher than the predominant frequency of the structure, mainly leaving frequency components that are close to or lower than the predominant frequency of the structure. The most critical frequency components of the input motion are those that are close to the predominant frequency of the structure and to the natural frequency of the floor isolation.

Equation of motion for floor isolation system

Considering the isolated floor and the equipment on the top as a rigid body, the floor isolation can be represented with an SDOF model:

(1)\ddot{x}+\mathrm{2}\zeta {\omega }_{}\dot{x}+{\omega }^{\mathrm{2}}x+f_{}/m+u/m=-{\ddot{x}}_{},

where x is the relative displacement of the floor isolation system with respect to the structure floor, z is the viscous damping ratio, w_n is the natural frequency of the floor isolation system, f_p is the friction on the contact surface of the rolling pendulum, u is the control force from the damper, {\ddot{x}}_{} is the input acceleration, and m is the total mass of the floor isolation system. In this study, the viscous damping ratio z of the designed floor isolation system is small enough to be assumed as zero.

Combining the friction force and the damper control force gives:

(2)\ddot{x}+\mathrm{2}\zeta {\omega }_{}\dot{x}+{\omega }^{\mathrm{2}}x+u^{*}/m=-{\ddot{x}}_{},

where u* = f_p + u.

Equation (2) can be transferred into the state space form:

(3)\dot{X}=AX+B{u}^{*}+H{\ddot{x}}_{},

where

X=\left[\begin{array}{l}x\\ \dot{x}\end{array}\right],A=\left[\begin{array}{ll}\mathrm{0}& \mathrm{1}\\ -{\omega }_n^{\mathrm{2}}& -\text{2}\zeta {\omega }_{}\end{array}\right],B=\left[\begin{array}{l}\mathrm{0}\\ -\mathrm{1}/m\end{array}\right],H=\left[\begin{array}{l}\mathrm{0}\\ -\mathrm{1}\end{array}\right]

Design target

To protect equipment from damage, the acceleration of the floor isolation must be reduced. However, as space for the floor isolation system is limited, small clearance distances between the floor isolation system and surrounding walls are desirable to maximize the usable place on the isolated floor.

Figure 2 shows the transfer functions from the input acceleration to the floor acceleration and displacement for a passive control system with viscous damping. Two different damping ratios (0 and 40%) were adopted in the simulation. The floor acceleration transfer function shows that when the ratio w/w_n between the input acceleration frequency and the natural frequency of the system is larger than \sqrt{\mathrm{2}}, the smaller is the damping, the smaller is the acceleration; when w/w_n is smaller than \sqrt{\mathrm{2}}, the larger is the damping, the smaller is the acceleration. The floor displacement transfer function shows that larger damping is always more effective in reducing displacement, but the difference in the displacement response using different damping ratios is negligible for high frequency inputs (w/w_n>2).

As indicated in Section 2.2, the most critical frequency components of the input motion are the high frequency components that are close to the predominant frequency of the structure because of the high amplitude, and those low frequency components that are close to the natural frequency of the floor isolation because of the resonance with the floor isolation. For high frequency input, low damping is preferred, while high damping is preferred for low frequency input. An optimal control design for the floor isolation system should satisfy both of these requirements. This study tries to design a system that has similar responses as a passive system with very low damping for the input motion with frequencies close to the predominant frequency of the structure. However, when the input motion is dominated by frequency components close to the natural frequency of the floor isolation system, the designed system uses higher damping to prevent resonance.

Semi-active control for a floor isolation system using an H_∞ algorithm

To design a system with the damping that can be varied with respect to the input motion, a semi-active control is adopted. The efficiency of the semi-active control depends on the control algorithm, and the information about the controlled system and input motion used in the control design. A lack of knowledge about the incoming earthquake will make it difficult to consider the input motion characteristics and lessen the efficiency of control. However, the unique configuration of the floor isolation system enables the designer to know the basic features of the input motion to the floor isolation, as described in Section 2.2. This feature can be described using a frequency domain model, and the model can be implemented in a frequency control method [12]. A conventional H_∞ control algorithm [10-12] is adopted because of its efficiency in implementing the input motion model in the frequency domain.

Formulation of H_∞ control

A frequency dependent filter W₁ was designed to characterize the input motion {\ddot{x}}_{}, considering the characteristics of the input motion to the floor isolation system, i.e., mainly the frequency components of the ground motion that are close to or lower than the predominant frequency of the structure, remain after the filtering effects of the structure. Filter W₁ can be considered as the transfer function, with the white noise w as the input and the real earthquake excitation as the output. In addition, a filter W₂ was also designed to regulate the control on acceleration and displacement:

(4){\ddot{x}}_{}=W_{\mathrm{1}}w,

(5)z=W_{\mathrm{2}}{C}_{z}X+D_z{u}^{*},

where

C_z=\left[\begin{array}{ll}-{\omega }_{}^{}& -\mathrm{2}\zeta {\omega }_{}\\ \mathrm{1}& \mathrm{0}\end{array}\right],D_z=\left[\begin{array}{l}-\mathrm{1}/m\\ \mathrm{0}\end{array}\right].

By appending the filters to the floor isolation system, it results in a typically higher order augmented system. Figure 3 shows the control diagram of the augmented floor isolation system using H_∞ control. BoldItalic is the designed H_∞ controller. In this system, the measurement output y for the controller includes the displacement and velocity, i.e., BoldItalic = BoldItalic.

With the designed controller BoldItalic, the control force u* can be obtained as:

(6)u^{*}=Ky.

The central idea of H_∞ control is to design a controller BoldItalicthat minimizes the ∞ norm of the transfer function of the augmented floor isolation system from the input w to the regulated response BoldItalic, H_{zw}:

(7)

where sup denotes the supremum, \overline{\sigma } stands for the maximum singular value of the transfer function, and γ is a positive bound for the norm. The transfer function can be expressed as

(8)H_{zw}=(W_{\mathrm{2}}{C}_{}+D_{}K){(sI-A-BK)}^{-\mathrm{1}}H{W}_{\mathrm{1}},

where s represents the Laplace transform variable hereafter. A magnetic-rheological (MR) damper [13,14] was adopted as the semi-active control device in this study, to produce the desired control force u^* that was calculated from Eq. (6).

Design of filters W₁ and W₂

The order of the augmented system shown in Fig. 3 depends on the order of the additional filters. Generally, it is easier to implement control for a lower order system than a higher order system. It is therefore desirable to employ low order models for the two filters.

Filter W₁ can be designed with different shapes [15-17] depending on the frequency characteristics of the input motion. Considering the characteristics of the input motion for the floor isolation system, the frequency component close to or lower than the predominant frequency of the structure is most influential. A new shaping filter is designed

(9)W_{\mathrm{1}}={\displaystyle \frac{\mathrm{4}{\zeta }_{}{\omega }_{}s+\chi {\omega }_{}^{\mathrm{2}}}{s^{\mathrm{2}}+\mathrm{2}{\zeta }_{}{\omega }_{}s+{\omega }_{}^{\mathrm{2}}}},

where {\zeta }_{} = 0.3 [16,17].The filter is designed to have the frequency peak corresponding to the predominant frequency of the structure ω_p, i.e., ω_f = ω_p. The χ parameter adjusts the control effort on different frequency bands. When the χ parameter is larger, the filter has a larger power spectrum density (PSD) at a lower frequency region. With this setting, more control effort is shifted to lower frequency region.

Figure 4 shows the PSD of W₁ for two different χ values (0 and 0.3). ω_f is chosen as 13.2 rad/s (2.1 Hz). The PSD drops off sharply on the high frequency side, indicating that the frequency content of the input motion higher than the predominant frequency of the structure is significantly filtered out. W₁ with χ = 0 has a much lower PSD in the low frequency region than when using χ = 0.3.The setting χ = 0 is found not able to cover the possible low frequency components around the natural frequency of the floor isolation. By increasing the value of χ, more control effort is shifted to the low frequency region and it will achieve the better performance for input motions with low frequency (long period).

When more control effort is placed on the reduction of displacement in the low frequency region, relative importance on the reduction of acceleration in the high frequency region becomes low. Therefore, χ is also an indicator for the adjustment of control effort on the acceleration and displacement.

The state space expression of W₁ is shown in Eq. (10):

(10)\begin{array}{l}{\dot{X}}_{W\mathrm{1}}=A_{W\mathrm{1}}{X}_{W\mathrm{1}}+B_{W\mathrm{1}}w,\\ {\ddot{x}}_{}=C_{W\mathrm{1}}{X}_{W\mathrm{1}},\end{array}

where BoldItalic_W1 is a state variable and w is the white noise excitation. BoldItalic_W1, BoldItalic_W1 and BoldItalic_W1 are system matrices:

(11)A_w=\left[\begin{array}{ll}\mathrm{0}& \mathrm{1}\\ -{\omega }_{}^{\mathrm{2}}& -\mathrm{2}{\zeta }_{}{\omega }_{}\end{array}\right],B_w=\left[\begin{array}{l}\mathrm{0}\\ \mathrm{1}\end{array}\right],C_w=\left[\begin{array}{ll}\chi {\omega }_{}^{\mathrm{2}}& \mathrm{4}{\zeta }_{}{\omega }_{}\end{array}\right].

Filter W₂ is designed to weight the control targets as

(12)W_{\mathrm{2}}=\left[\begin{array}{ll}\alpha & \mathrm{0}\\ \mathrm{0}& (\mathrm{1}-\alpha )W_{\mathrm{3}}\end{array}\right],

where a is a scalar weighting corresponding to the absolute acceleration of the floor isolation excluding the portion caused by the damper force u^*/m, as shown in Eq. (5). (1- a)W₃ is a frequency dependent filter corresponding to the displacement. W₃ is designed as a first order filter with a roll off frequency of 0.4 Hz. It is used to regulate the displacement under motions with frequencies close to or lower than the natural frequency (0.33 Hz) of the floor isolation system.

(13)W_{\mathrm{3}}={\displaystyle \frac{\mathrm{0.4}\times \mathrm{2}\pi }{s+\mathrm{0.4}\times \mathrm{2}\pi }}.

Transfer function analysis

Figure 5 shows the transfer functions of the floor isolation system from input acceleration to floor acceleration and displacement. Two passive control results with damping ratios of 0 and 0.4, and three H_∞ control results are presented. For H_∞ control, ω_f is chosen as 13.2 rad/s (2.1 Hz) for filter W₁. By simulation, the parameters χ and a in filter W₁ and W₂ are tuned to be 0.3 and 0.98, respectively. This allows the H_∞ control with the two filters to have a similar acceleration and displacement with the passive control using a damping of zero, at the frequency that is close to the predominant frequency ω_p of the structure; while the response around the natural frequency of the floor isolation is close to or lower than those obtained for passive control using high damping of 0.4.The figure also shows that using two filters more effectively reduces acceleration rather than using either W₁ or W₂, or without using any of them.

Figure 6 shows the transfer function of the designed H_∞ controller BoldItalic (it has two inputs, i.e., displacement and velocity of the floor isolation). It is notable that the magnitude of the transfer function from the input displacement to the output force is relatively constant, and the one from the input velocity to the output force varies according to the input velocity frequency. When the input velocity dominates with the frequency close to the natural frequency of the floor isolation, the force magnitude is large, indicating a large damping in the system. When the input velocity dominates with a frequency close to the predominant frequency of the structure (2.1 Hz), the force magnitude is low, which indicates a low damping in the system.

Shaking table test for the floor isolation system

Test setup

To examine the behavior of the semi-active floor isolation system with H_∞ control, a series of shaking table tests were conducted. Figure 7(a) shows the test setup of the floor isolation specimen with a plan of 2.5 m by 2.5 m. Four sets of unidirectional rolling pendulum isolators were installed under the floor isolation system. The friction coefficient of the isolators was 0.01, and the identified natural period of the pendulum was 3.0 s.

A steel frame was used to represent the isolated floor. The MR damper was connected to the shaking table and the steel frame to supply the semi-active control force. The total weight of the setup, including the steel frame and rolling pendulum isolators, was 62.5 kN. A displacement restrainer that represented the surrounding wall was installed under the steel frame to prevent the floor isolation system from moving over 200 mm, which was the displacement limit of the MR damper.

A photograph of the control system setup is shown in Fig. 7(b), including a computer system equipped with a DSP chip, A/D and D/A converters for data transfer, and a current driver (see Fig. 7(c)) to supply power to the MR damper by the calculated control signal. Displacement and velocity were used as feedback signals for H_∞ control. The displacement was measured with a magnetostrictive displacement transducer mounted on the MR damper. The velocity was determined by passing the displacement data through a second order Butterworth filter with a cut-off frequency of 30 Hz.

A passive floor isolation system using an oil damper was also tested. The oil damper was designed to have the same 10 kN load capacity as the MR damper (maximum current 3 A) at a velocity of 1 m/s. The damper force f_oil was proportional to the velocity of the oil damper\dot{x}, i.e., f_{}=c\dot{x}(c = 10 kN·s/m). For the floor isolation system using the oil damper in this study, with a weight of 62.5 kN and a natural period of 3.0 s, the damping ratio of the floor isolation system was 38%.

MR damper properties

The MR damper had a design force capacity of 10 kN and a±200 mm stroke. Current was used as the control signal, and the allowable current ranged from 0 to 3 A. Figure 8 shows the force-displacement and force-velocity relationships when the MR damper was subjected to 1.28 Hz sinusoid motion with a maximum displacement of 100 mm (maximum velocity of 0.8 m/s), for seven different levels of current. As observed from Fig. 8(b), the force of the MR damper is a function of the input current and velocity.

The MR damper is an intrinsically energy dissipation device and cannot add mechanical energy to the controlled system. As such, part of the desired force calculated by the H_∞ control cannot be achieved by the MR damper [18]. To achieve a force as close to the calculated value as possible, a PI controller [19] was designed to calculate an appropriate current for the MR damper.

Figure 9 shows how the H_∞ control using the MR damper and the PI controller were achieved in the shaking table test. With the feedback signals BoldItalic, the control system calculated the desired force u^* by the H_∞ control as shown in Eq. (6). The information of the desired force u^* was fed to the PI controller. By comparing with the measured actual force u of the MR damper, the PI controller calculates the current (I in Fig. 9) needed for the MR damper to track the desired force.

Input motions

To examine the performance of the floor isolation system configured as shown in Fig. 1, three recorded floor responses, JMA_R, SAN_R and ELC_R, were adopted. They were obtained from a five-story full-scale RC structure tested using the E-Defense shaking Table [7], which is the largest shaking table in the world. Table 1 lists the basic information of the adopted motions including the predominant frequency w_p of the structure when each ground motion was applied. The structure was tested many times for different motions. The damages during the tests resulted in the changes of the dominant frequency. Figure 10 shows FFT results of the acceleration records. From the figure, frequencies higher than the predominant frequency of the structure were significantly filtered out.

Motions JMA_R, and ELC_R were short period motions, which had high amplitudes and relatively high dominant frequencies that are close to the predominant frequency of the structure, respectively. Motion SAN_R was a long period motion, dominating at 0.35 Hz and the frequency content at higher frequency band had significantly lower amplitude. The short period and long period motions were used to examine the capacity of H_∞ control when dealing with motions that have different frequency characteristics.

Another ground motion, ELC, which generated the floor response of ELC_R, was also adopted. Unlike ELC_R whose frequency component was mainly distributed around the predominant frequency w_p, ELC had a much wider frequency band, ranging from 0.5 to 5 Hz. This motion was used to compare the effectiveness of H_∞ control in controlling the input motion of the floor response and ground motion.

Test results

A series of shaking table tests were performed to validate the performance of the floor isolation system using different control strategies, including semi-active control with H_∞ algorithm and passive control. The performances of different controls were evaluated based on two indices; J₁ (m/s²) and J₂ (mm). They respectively represent the peak acceleration and the peak displacement responses of the floor isolation. The acceleration response of the floor isolation system significantly influences the equipment response. The peak displacement response is an important index for evaluating whether the control is effective in limiting displacement. The test results are shown in Table 2 and compared graphically in Fig. 11.

Effectiveness of H_∞ control

With the control target for the floor isolation system discussed in Section 2.4, the parameters χ and a in filter W₁ and W₂were tuned to 0.3 and 0.98, respectively. To examine the effect of different c values for input filter W₁ on balancing the reduction of acceleration and displacement, a c value of 0.4 was also used.

When both the two filters W₁ and W₂ were adopted and the parameter w_f was assigned as w_p, the H_∞ control could effectively reduce the acceleration and displacement. Using c = 0.3, the acceleration was reduced to 8% of the input for the short period motions JMA_R and ELC_R, and 24% for the long period motion SAN_R. The control was more effective on reducing acceleration at c = 0.3 than at c = 0.4 for the short period motions JMA_R and ELC_R, as shown in Table 2 and Fig. 11. This was primarily because the input filter W₁ at c = 0.3 had a relatively lower PSD in the low frequency band, and a relatively higher PSD in the frequency band close to the predominant frequency of the structure. This is also the reason to the lower efficiency in reducing the displacement for the long period motion SAN_R at lower c values.

The input filter W₁ for the H_∞ control was important for controlling the floor response. Without filter W₁, H_∞ control was effective in reducing acceleration for the long period SAN_R motion, but was ineffective for the short period JMA_R motion. This was attributed to more focus given to controlling low frequency (long period) components in the input motion, which resulted in larger damping in the system. In this condition, the H_∞ control aimed to reduce the response of the input motions with frequencies close to the natural frequency of the floor isolation. Figure 12 shows the velocity versus MR damper force relationships. It can be observed that the force without W₁ was significantly larger than that with W₁, which indicates a larger damping in the system.

The long period motion SAN_R dominated at 0.35 Hz and was close to the 0.33 Hz natural frequency of the floor isolation system, the larger control force without W₁ benefited the reduction of floor response in this case. However, when the short period motion JMA_R excited the floor isolation, the larger control force (larger damping) was against the control strategy as illustrated in Fig. 2, i.e., a smaller damping was needed for the short period motion. The acceleration and displacement without W₁ increased respectively by 135% and 37%, compared to that with W₁ (c = 0.3). Without informing the controller of the input motion characteristics, the control used the same control strategy, i.e., larger damping force in the system to control the floor isolation response.

Comparison between passive control and semi-active control with H_∞ algorithm

Compared with the passive control using oil damper, H_∞ control with parameters w_f = w_p was found to be more effective. For the short period motion JMA_R, the ratio of floor acceleration using H_∞ control (c = 0.3) to that using passive control was 46%; in the long period motion SAN_R, this ratio was only 35%. In addition, H_∞ control further reduced displacement by 20% under the long period motion SAN_R.

As illustrated in Section 2.4, to further reduce the floor acceleration for the short period motion JMA_R, it is necessary to decrease the damping level for the passive control. However, this increases both the acceleration and displacement responses under the long period motion SAN_R. Therefore, it is difficult to use passive control to deal with both short period motions having frequency components close to the predominant frequency of the structure, as well as long period motions with frequency components close to the natural frequency of the floor isolation system.

Semi-active control with H_∞ algorithm was more advantageous than passive control for dealing with both short and long period motions, thanks to its capacity to vary the control strategy with respect to the frequency characteristics of the input motion.

Selection of parameter w_f for filter W₁

Figure 13 shows a comparison of the normalized maximum floor acceleration by the maximum input acceleration for different testing conditions. Two different w_f values were assigned to filter W₁ under JMA_R motion. One of the w_f values was the same as the predominant frequency of the structure w_p (8.8 rad/s), and the other one was 13.2 rad/s. The comparison shows that when w_f was adopted as w_p, H_∞ control was more effective than when w_f was at the higher value.

Comparing the results between the input motion ELC (ground motion) and ELC_R (roof motion) shows that the designed H_∞ control was more effective for ELC_R motion in terms of the normalized acceleration. The ratio between the floor acceleration and input motion were 22% and 8%for ELC and ELC_R respectively. The differences were attributed to the different input motion characteristics. Motion ELC_R was dominated at the frequency that was close to the predominant frequency of the structure. When the H_∞ control was designed to have w_f = w_p, the filter W₁ correctly captured the frequency characteristics of the input motion, and H_∞ control effectively reduced the floor response. However, the frequency of motion ELC was distributed over a wider frequency band and was not able to be captured by the input shaping filter. Therefore, when the H_∞ control was used for the two motions, it was more effective for the ELC_R motion. This shows the effectiveness of using H_∞ control for the floor isolation.

Conclusions

A floor isolation system installed in a fixed base structure was designed to protect equipment in a single floor or room. To mitigate the acceleration of the floor isolation and suppress its displacement particularly under a long period motion, a semi-active control was adopted with an H_∞ algorithm using a newly developed input shaping filter. The H_∞ control was designed through the transfer function analysis and validated by a series of shake table tests for a semi-active floor isolation system equipped with rolling pendulum isolators and an MR damper. Passive control using an oil damper was also conducted for comparison. The following findings are notable from the study.

1) The input motion to the floor isolation is filtered from the ground motion due to the vibration characteristics of the structure, leaving mainly frequency components lower than the predominant frequency w_p of the structure. Such characteristics are utilized in the H_∞ control which makes the control more effective.

2) A new input shaping filter is designed to consider the frequency characteristics of the input motion to the floor isolation. An H_∞ control is designed with the shaping filter to deal with both short and long period motions. Transfer function analysis shows that the H_∞ control results in similar acceleration and displacement with the passive control using viscous damping ratio of zero at frequencies close to the predominant frequency w_p of the structure, while the acceleration and displacement is close to or lower than the passive control using a viscous damping ratio of 0.4 around the natural frequency of the floor isolation.

3) The advantage of H_∞ control designed with the input shaping filter over passive control is validated in the test. The H_∞ control can reduce the floor responses more effectively for motions with dominant frequencies close to both the predominant frequency of the structure (short period) and the natural frequency of the floor isolation (long period).

4) The input motion filter is important in the design of H_∞ control. Without the input filter, control efforts tend to be shifted toward low frequency input motion. The test results show that the control over the motion which has a dominant frequency close to the natural period of the floor isolation is effective. However, the control cannot suppress acceleration under motions with the dominant frequency close to the predominant frequency of the structure.

5) It is more effective to control the acceleration of the floor isolation under the floor motion compared with the ground motion which results the floor motion. The knowledge on the vibration characteristics of the floor motion enhances the effectiveness of H_∞ control.

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