Structures in Seismic Zones. J. Georey Chase 2. This paper presents the ndings of a study devoted to a comparison of the eectiveness

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1 Comparison of LQR and H1 Algorithms for Vibration Control of Structures in Seismic Zones Abstract H. Allison Smith 1 (Assoc. Member) J. Georey Chase 2 This paper presents the ndings of a study devoted to a comparison of the eectiveness of Linear Quadratic Regulator (LQR) and H1 control for structures subjected to earthquake loadings. Research in the eld of active structural control has received much attention over the past few years, with the majority of eorts focusing on formulation and application of the LQR-based control algorithms. Because of various drawbacks associated with the LQR algorithms, such as inability to account for time variant structural properties and the time history of the loading, other control algorithms such as H1 have been investigated as viable alternative control algorithms for application to civil structures. In this study, a comparison is made between the LQR and H1 algorithms based on application of each algorithm to an actively controlled 33-story structure. This structure, located in Tokyo, is equipped with two rooftop active mass dampers (AMD's) capable of eectively increasing damping and controlling vibratory behavior in four vibration modes. Comparisons of the control algorithms are performed on shear building representations (ve degrees-of-freedom) of the structure. Both actuator eort and vibration reduction under small to medium earthquake excitations are considered in this discussion. Additional studies are performed to investigate the robustness of the control scheme considering non-linear actuator saturation eects and time varying uncertainties in the structural parameters. Introduction Robust H1 state feedback control systems account for modelling errors and time varying structural parameters while limiting the worst case rms energy transfer from the disturbance input to the regulated outputs [Smith et. al., 1994]; These controllers also are capable of guaranteeing stability for closed loop systems with time varying structural uncertainty in the presence of non-linear actuator saturation eects [Chase et. al., 1995a,b]. Robust H1 static output feedback controllers have been investigated by Schmitendorf et. al. [1994] for structural systems, and Smith et. al. [1995] discussed applications to structural acoustic problems. This study compares the eectiveness of robust H1 state feedback controllers accounting for actuator saturation to that of typically designed LQR state feedback controllers. These dierent control systems are 1 Professor, Dept. of Civil Engineering, Stanford University, Stanford, CA , U.S.A. 2 Graduate Student 1

2 applied to an existing actively controlled structure to create a realistic comparison of their capabilities. The 33-story Riverside Sumida Central Tower is a moment resisting, steel frame structure designed and constructed by Obayashi Corporation and located in Tokyo, Japan (see Figure 1). The building is tted with two active mass dampers (AMD's), located on the roof, for reduction of structural response due to small earthquake inputs (10 to 50 cm=sec 2 peak ground acceleration) and wind loads. Consequently, the size of the AMD's is small (0.06%) compared to the total mass of the structure. However, although signicant peak vibration control due to full scale seismic excitations is not possible with this actuator, the eectiveness of various control algorithms can be tested using this structure-control model. The design, development, and testing of this system have been reported extensively by Suzuki et. al. [1994], and Watanabe et. al. [1992]. Figure 1: Riverside Sumida Central Tower, Tokyo This research uses a model of this structure developed by Suzuki et. al. [1990] for a single lateral direction. The model was derived from modal data collected from ve instrumented locations on the structure during full scale testing [Suzuki et. al., 1994], resulting in a ve degree-of-freedom (d.o.f.) model representative of the rst ve lateral modes of the structure. The rst two fundamental frequencies of the structure are quite low (T 1 2:9 seconds and T 2 1:0 seconds), placing the structure's fundamental modes in a region of larger peaks on the displacement response spectra employed by earthquake engineers for seismic risk assessment. In addition, the rst mode is lightly damped (0:85% of critical) creating an easily excited structure with limited attenuation properties. Thus, the maintenance of stability and the importance of controlling the response in the presence of time varying parametric uncertainties and actuator saturation presents a realistic and dicult control design problem. Complete 2

3 modal frequency and damping data for this structure are presented in Suzuki et. al. [1994]. The following sections present the complete case study comparing these two control algorithms. Details on how the H1 controllers were designed are available in Chase et. al. [1995]. The nal section presents the conclusions of this study. H1 and LQR Control of the Sumida Tower The state space system considered for this linear, time-varying, uncertain structural system is given by: _x(t) = A(t)x(t) + B 1 u(t) + B 2 w(t) (1) z(t) = C 2 x(t) (2) where x(t) is the state vector, u(t) is the vector of control inputs, w(t) is the vector of disturbance inputs, and z(t) is the vector of regulated outputs; B 1, B 2, C 1, and C 2 are the associated matrices mapping these vectors into the state space, while A(t) is the time varying plant matrix. The plant matrix in equation (1) is time varying to account for the time varying uncertainties of the structural parameters. This plant matrix may be more explicitly represented as the sum of a nominal, constant, matrix and a matrix of time varying uncertainties, A(t) = A o + DF(t)E (3) where D and E are known constant matrices, and F(t) is a matrix of unknown functions, which are assumed to be bounded over time. Because of the limited mass of the installed AMD, the objective function for the controller design optimization problem is chosen as the minimization of peak actuator eort. In addition, this AMD saturates at an acceleration demand of 280cm=sec 2 for a maximum input force of 8:7 metric tons. Therefore, minimizing the peak actuator eort for a given set of performance constraints maximizes the eect of this controller on the structural response by minimizing the amount of actuator saturation that occurs for a given disturbance input. The design of H1 controllers requires the selection of a set of regulated outputs z(t). In this study, these regulated outputs are chosen as the interstory drift displacements between modeled degrees-of-freedom. The goal is to create a controller which has a minimum objective and satises the constraint that the closed loop system possess an attenuation constant for which the following H1 norm condition is valid, kt zw k1 = sup kz(t)k 2 (4) w(t) kw(t)k 2 where sup is dened as the supremum over all w(t), is a positive, scalar attenuation constant, and both w(t) and z(t) are assumed to be bounded. The resulting controller has a closed loop attenuation constant CL which is less than, or equal to, in Equation (4) and has minimized the objective function selected. Stable systems with values of less than 1:0 are referred to as non-expansive. For a given set of regulated outputs z(t), the larger the value of the attenuation constant, the larger the energy transferred from the input excitation to the regulated outputs, in the worst case. The open loop, nominal attenuation constant for the Sumida Tower is found to be OL?nom = 10:8 representing an expansive system transfer from the seismic disturbance inputs to the regulated outputs. 3

4 The uncertainty selected for this analysis is a 17% variation of the frequency of the rst structural mode. This variation corresponds to the width of the rst mode peak on the singular value plot for this structural system's transfer functions between w(t) and z(t). The uncertainty is modeled using Equation (3) and the structure's modal data. This set of possible variations is then accounted for, along with actuator saturation eects, in the creation of the H1 controllers using the methods described in Chase et. al. [1995]. Only the rst mode is modeled as uncertain because this mode dominates the value of the H1 norm as well as the seismic response of this tall structure. Control Design Given the control objectives mentioned previously several robust H1 state feedback controllers were created by varying the value of CL desired over the range [0:4 11:0]. Values greater than 11:0 were not selected since that would guarantee an attenuation constant for the closed loop system greater than that of the nominal open loop model ( OL?nom). Values of CL less than 0:4 did not yield feasible controllers when actuator saturation was considered. From all of these controllers a single controller was selected with a design value of CL = 5:0. For comparison with this robust H1 controller, a linear quadratic regulator (LQR) controller is designed using the typical approach employed in prior civil engineering active control literature [Soong, 1990]. This approach entails the use of linear time invariant (LTI) models and the solution of an algebraic Riccati equation (ARE) to obtain a set of state feedback gains. These state feedback gains are designed to optimally minimize the quadratic performance objective J given as J = Z T 0 [x(t) T ^Qx(t) + u(t) T Ru(t)]dt (5) where ^Q and R are known, constant weighting matrices. The relative values of these weighting matrices determines the relative emphasis between control economy, or minimal control input, and response attenuation, or minimal response. Hence, the actual values used for each matrix are of lesser note than the relative value between them. When ^Q is much larger valued than R, the emphasis is on response attenuation. This case is referred to as the \cheap control" case since there is relatively little emphasis on minimizing control eort. Conversely, when ^Q is much smaller than R, the emphasis is on minimizing control eort and is referred to as the \expensive control" case. The typical LQR design approach, employed in designing control for civil structures, selects these weighting matrices based on obtaining a controller that does not saturate for a series of simulation inputs. In this case, three scaled earthquake inputs are used since this AMD is designed primarily for attenuation of responses due to smaller ground motion inputs (PGA50.0 cm=sec 2 ). Specically, the matrix ^Q is chosen as identity and the scalar matrix R is modied, since it is the relative value of the weighting matrices that is important, until appropriate LTI simulation results are obtained. Simulation of Controllers There are two sets of simulations for each of three dierent earthquake inputs presented in this section. The rst set considers inputs for which the peak ground acceleration (PGA) has been scaled to 50cm=sec 2 for each input, resulting in an earthquake that produces only small responses in the structure's linear range such that a LTI model is sucient. The second set employs the unscaled full record which produces responses well outside the linear range, for which a time varying damage model 4

5 has been created to simulate a possible variation in eective stiness. Hence, the rst set of simulations examines the sensitivity of the nominal structure to the frequency content of the dierent earthquakes and illustrates the ability of the dierent controllers to control responses in the occupant comfort range. The second set examines the ability of the control system (controller plus actuator) to reduce peak response and attenuate transient response in the presence of extreme actuator saturation (and the resultant clipping) and time varying structural parameters. The primary advantage of the H1 controller versus the LQR controller described above is that the H1 controller guarantees stability and worst case performance in the presence of time varying uncertainty and actuator saturation. For the LQR controllers, the level of robustness when considering these non-linearities cannot be explicitly considered a priori and, instead, can only be estimated through simulation. In this study, open loop simulations are performed to ascertain what levels of attenuation are achieved for each control system. The three earthquakes chosen for these simulations are selected due to the large energy present in their displacement response spectra over the T = [1:0 3:0] second period range, increasing the ability to excite the rst two modes of the Tower. In this context \average" refers to the ATC-S 1 code specied design displacement response spectra. The ground motion records selected are: Sylmar Record - Northridge, CA PGA = 826cm=sec 2 Holtville Record- Imperial Valley, CA PGA = 246cm=sec 2 Tabas Record - Tabas, Iran PGA = 729cm=sec 2 Modeling errors and parametric uncertainty are simulated through a 17% reduction in the rst natural frequency as a function of the displacement of the rst modal coordinate. This specic frequency variation is intended to represent a possible increase in exibility due to damage while undergoing non-linear response. Such increases are often coupled with increases in damping, as noted in hysteresis loops where the slope is reduced (decreasing dynamic stiness) and grows from the linear line to a wider loop (increasing structural damping). In modeling only the increased exibility without any increase in structural damping, a conservative model is created for control design. For each simulation the total energy norm for all the story drift measurements (k z(t) k 2 ) and for the control input (k u(t) k 2 ) are recorded along with the peak magnitude story drifts (max j z(t) j) and the decay time (t D ). The decay time represents the time at which the response of the rst oor story drift becomes less than 0:1cm in magnitude. Tables 1 and 2 present this data for all the simulations performed. Note that the rst d.o.f. of this model represents the roof of the structure; hence, the peak magnitude story drifts as presented are ordered from the top of the structure to its base. As mentioned earlier, the AMD mounted on the Tower are intended for comfort control (i.e., control of vibration due to small wind loads that may make building occupants uncomfortable). Therefore, it is not surprising that the peak story drifts reported in Table 1 show that the AMD on this structure is too small to eectively reduce the peak structural responses generated by these scaled earthquake loadings. Given the large size of the Tower, it would be dicult to design and implement an AMD large enough to have signicant impact on these responses. Hence, only comfort issues, as reected by the rate of disturbance attenuation, are valid means of comparison for the dierent controllers. Note that none of the peak story drifts in Table 1 exceeds the 2.2 inch (5:6 cm) linear limit (dened using UBC), validating the decision to treat the structure as a LTI system. However, each of the controllers reduces the rms energy of the response, as reected in reduced response, k z(t) k 2 and decay time, t D, when compared to 5

6 Table 1: Time Invariant Simulation Results For Scaled Earthquake Inputs Controller Earthquake k z(t) k 2 k u(t) k 2 t D max j z(t) j Record (5th,...,1st story) Open Loop Tabas [ ] Sylmar [ ] H1 Holtville [ ] Tabas [ ] Sylmar [ ] Holtville [ ] LQR Tabas [ ] Sylmar [ ] Holtville [ ] the uncontrolled state. More importantly, the robust H1 controller performs better than the LQR controller; however, this is accomplished at the expense of more control energy. It is important to note that the LQR controller was designed such that actuator saturation, which occurs at 280cm=sec 2 for this AMD, does not occur for these scaled inputs. For this reason the rms energy of the control inputs k u(t) k 2 is lower for the LQR controller. However, since the robust H1 controller guarantees stability in the presence of actuator saturation, it is able to saturate the controller and use its maximum capacity to achieve better response results and shorter decay times. It is worth noting that the Holtville record has the worst frequency content with respect to exciting the nominal structure, as reected in the larger response values for this input. The low damping value for the rst mode of the structure is reected in the long decay times for the open loop simulations. For the unscaled (full magnitude) set of simulations presented in Table 2, both controllers saturated by signicant amounts, and their control inputs were clipped to the AMD's capacity of 280cm=sec 2. Table 2: Simulation Results For Full Records And Time Varying System Controller Earthquake k z(t) k 2 k u(t) k 2 t D max j z(t) j (cm) Record (5th,...,1st story Open Loop Tabas [ ] Sylmar [ ] H1 Holtville [ ] Tabas [ ] Sylmar [ ] Holtville [ ] LQR Tabas [ ] Sylmar [ ] Holtville [ ] As seen with the scaled input simulations, the decay time for the robust H1 controller is signicantly better than that for the LQR controller at the expense of a higher actuator eort. Using the increased exibility model, each of the three unscaled simulations caused a rapid 17% reduction in the structure's rst natural frequency. The larger gains for the H1 controller lead to a longer period of actuator sat- 6

7 uration. However, the resulting larger control input energy leads to a more rapid attenuation of the transient responses. Thus, the ability to guarantee stability in the presence of actuator saturation allows the designer to employ these larger control gains with this robust H1 method. The traditional LQR design approach avoids explicit consideration of actuator saturation and, instead, bases controller design on avoiding this condition through simulation of maximal events. With such small capacity actuators, such an approach for these unscaled inputs would result in a LQR controller with signicantly worse performance than the LQR controller designed in this study. Conclusions This research presents a detailed case study comparing the eectiveness of robust H1 controllers which account for non-linear actuator saturation eects versus LQR controllers. These controllers are designed for the Riverside Sumida Central Tower in Tokyo, a 33-story steel framed structure equipped with an AMD control system. The control objective for the constructed structure is the reduction of vibration which may cause occupant discomfort; thus, the AMD mass is small (0:06%) compared to the overall structural mass. Included in this investigation is the use of an uncertainty model to represent potential variation of the structural frequencies. For all three earthquake records simulated, the robust H1 controllers are able to more rapidly attenuate the transient response, as measured in the decay time t D. Neither the H1 controller or the LQR controller are unable to signicantly reduce peak responses due to the small capacity of the AMD used. For small inputs, the AMD does have signicant impact on occupant comfort by rapidly damping out transient motion for this lightly damped structure. In this case, the performance of the H1 controllers is superior to the LQR controller. Finally, the LQR controller performed very well for the full record case in the face of structural parameter variation and extensive actuator saturation, verifying the typical robustness assumed of LQR designed controllers. However, the advantage of the H1 controllers over the typically designed LQR controller, beyond improved performance, is the ability to bound the worst case performance and, most importantly to explicitly guarantee this robustness as part of the design process. In summary, the robust H1 controllers developed in this investigation bound the worst case performance of the system and can explicitly consider additional performance objectives, uncertainties, and constraints. However, due to the relative size of the AMD, this controller was unable to aect the peak structural response, serving only to increase the attenuation of the structural response versus the open loop and LQR cases. Acknowledgements This research was supported in part by the National Science Foundation Grant number BCS , and the Obayashi Corporation. References 1. Chase, J. G. and Smith, H. A. (1995a). \Robust H1 Control Considering Actuator Saturation Part 1: Theory," (submitted to ASCE Journal of Engineering Mechanics). 2. Chase, J. G. and Smith, H. A. (1995b). \Robust H1 Control Considering Actuator Saturation Part 2: Applications," (submitted to ASCE Journal of Engineering Mechanics). 7

8 3. Schmitendorf, W. E., Kose, I. E., Jabbari, F., and Yang, J. N. (1994). \H1 Control Of Seismic Excited Buildings Using Direct Output Feedback," Proceedings of First World Conference on Structural Control, Los Angeles, California. 4. Smith, H. A. and Chase, J. G. (1994). \Robust Disturbance Rejection Using H1 Control For Civil Structures," Proceedings of First World Conference on Structural Control, Los Angeles, California. 5. Smith, H. A., Breneman, S. E., and Chase, J. G. (1995). \A Computational Approach For H1 Control Of The Exterior Structural-Acoustic Problem," Proceedings of the 1995 ASME International Congress, San Francisco, California. (in press) 6. Suzuki, T., Kageyama, M., Nohata, A., Seki, M., Teramura, A., and Takeda, T. (1990). \Active Vibration Control For High-Rise Buildings Using Dynamic Vibration Absorber Driven By Servo-Motor," Proceedings of the U.S. National Workshop on Structural Control Research, Los Angeles, California. 7. Suzuki, T., Kageyama, M., Nobata, A., Inaba, S., and Yoshida, O. (1994). \Active Vibration Control System Installed In A High-Rise Building," Proceedings of First World Conference on Structural Control, Los Angeles, California. 8. Watanabe, T., Yoshida, K., Shimogo, T., and Suzuki, T. (1992). \Reduced-order Active Vibration Control For High-rise Buildings," Proceedings of the First International Conference on Motion and Vibration Control (MOVIC), Yokohama, Japan. 8

9 KEY WORDS: State Feedback Control Structural Dynamics H1 Control LQR Control Earthquake Engineering Linear Time Varying Systems Robust Control Control Design 9

y 1 y 2 x 2 x 1 θ 1 θ 4 θ 2 θ 3

y 1 y 2 x 2 x 1 θ 1 θ 4 θ 2 θ 3 EVALUATING THE EFFECTIVENESS OF STRUCTURAL CONTROL WITHIN THE CONTEXT OF PERFORMANCE-BASED ENGINEERING Luciana R. Barroso 1, Scott E. Breneman 1, and H. Allison Smith ABSTRACT A preliminary study is presented