Predictive Optimal Linear Control of Inelastic Structures during Earthquake. Part II

The predictive optimal linear control (POLC) algorithm derived in the companion paper is extended to analyze the controlled responses of inelastic structures by incorporating the force analogy method (FAM). While POLC is very effective in compensating for the negative effect of time delay for elastic structures, the FAM is very efficient in performing the inelastic analysis. Different from conventional inelastic analysis methods of changing stiffness, the FAM analyzes inelastic structures by varying the structural displacement field, and therefore the state transition matrix needs to be computed only once. This greatly simplifies the computation and makes inelastic analysis readily applicable to the POLC algorithm. Numerical simulation is performed on a single-degree-of-freedom system to demonstrate the applicability of the POLC algorithm. Results show that the proposed control algorithm has feasibility for any inelastic structures for various control gains. Even though the control efficiency deteriorates with increase in time delay magnitudes, POLC maintains structural stability over a relatively wide range of time delay magnitudes. Finally, a computer model of a six-story moment-resisting steel frame is analyzed to show that POLC has a good control result on real inelastic structures, particularly in reducing the structural damages experienced during the earthquake.     

Model Predictive Control of Structures under Earthquakes using Acceleration Feedback

This paper presents a general formulation of the model predictive control (MPC) scheme with special reference to acceleration feedback in structural control under earthquakes. The MPC scheme is based on a prediction model of the system response to obtain the control action by minimizing an objective function. Optimization objectives include minimization of the difference between the predicted and desired response trajectories, and of the control effort subject to certain constraints. The effectiveness of MPC has been demonstrated to be equivalent to the optimal control. In this study, the prediction model is formulated using a feedback loop containing acceleration measurements from various locations in the structure. The state observer utilizes the Kalman-Bucy filter to estimate the states of the system from the acceleration feedback. Examples of single-story and three-story buildings equipped with control devices are used to demonstrate the effectiveness of the MPC scheme based on acceleration feedback. Both buildings are analyzed using an active tendon control device and an active mass damper (AMD). A two-story building with an AMD is used to experimentally validate the numerical control scheme. The results demonstrate the effectiveness of the MPC scheme using acceleration feedback. The acceleration feedback framework developed in this paper should serve as a building block for future extensions of MPC in capturing and benefiting from the attractive features of MPC, i.e., computational expediency, real-time applications, intrinsic compensation for time delays, and treatment of constraints, for implementation in civil structures.     

Earthquake Response and Energy Evaluation of Inelastic Structures

A computational method is derived to characterize the energy in inelastic structures and the transfer among various energy forms over the duration of an earthquake. This computational method is based on the force analogy method, which uses a change in displacement field to represent the inelastic behavior of structure instead of the traditional method of changing stiffness. The evaluation of plastic energy due to inelastic deformation in the structure becomes very simple using the force analogy method, where the accumulation of plastic energy due to plastic rotations is exactly equal to the elastic moment multiplied by the change in plastic rotations. In addition, this plastic energy formula can be used for any material with predefined stress–strain relationship, and therefore the transfer of energy among various forms can be calculated at any specific time. Once the energy equation is derived, numerical analyses are performed on a single degree of freedom system to study the characteristics of energy transfer. This is then extended to study the transfer of energy among various forms in a multidegree of freedom system. These two studies show that the analytically derived equation for plastic energy is accurate in studying the structural energy response due to earthquake excitations.     

Earthquake Response of Elastic Single-Degree-of-Freedom Systems with Nonlinear Viscoelastic Dampers

Investigated are the steady-forced and earthquake responses of single-degree-of-freedom (SDF) systems with a nonlinear viscoelastic damper (VED), which consists of a nonlinear fluid viscous damper (FVD) connected in series to a linear elastic bracing element (chevron or inverted V-shaped braces). For a wide range of bracing stiffness, nonlinear dampers are advantageous because they achieve essentially the same reduction in system responses but with a significantly reduced force. Damper nonlinearity has little influence on the structural response in the velocity-sensitive region of the spectrum even if the bracing is fairly flexible, but differences up to 16% were observed in other spectral regions. As expected, supplemental damping reduces structural response and the response reduction depends on the bracing stiffness, with this dependence varying with the spectral regions. For practical applications, a procedure is presented to estimate the design values of structural deformation, structural force, foundation shear, and damper force directly from the earthquake design (or response) spectrum. Finally, a procedure is presented to determine the damper and bracing properties necessary to limit the structural deformation to some design value or to the structural capacity.     

Optimal Design Strategy of Connected Control Method for Two Dynamically Similar Structures

Structural control of adjacent flexible structures has been shown to be effectively accomplished using the connected control method (CCM), which uses an auxiliary structure to provide a reaction force. A perceived constraint of the CCM is the ineffective control of structures with similar dynamics. However, the writers have presented a modified CCM mechanism in which dampers are connected between dynamically similar structures with a difference in connection height using cantilever structures. The research described in this paper extends the CCM to couple two dynamically similar structures and determines optimal stiffness and damping in the connector link. A two-degree-of-freedom building model is developed for the optimal design of the connected control devices. The optimal solutions of the connector stiffness and damping take into account the presence of fixed points in the systems transfer functions. Analytical results confirm the effectiveness of the proposed method and design. Experimental tests then verify the response reduction capabilities of the proposed optimal design.     

Optimal Control: Basis for Performance Comparison of Passive and Semiactive Isolation Systems

Passive damping in shock and vibration isolation systems reduces the deformation of the isolation system but can increase the acceleration sustained by the isolated object. Semiactive (i.e., controllable) damping systems offer a solution to the problem of increased vibration transmissibility at high frequencies. Semiactive damping is especially relevant to protecting acceleration-sensitive components to the effects of large impulsive earthquakes. In this paper, we compare three semiactive control policies, i.e., pseudonegative-stiffness control, continuous pseudoskyhook-damping control, and bang-bang pseudoskyhook-damping control, in terms of their effectiveness in addressing the deficiencies of passive isolation damping. In order to establish a performance goal for these suboptimal semiactive control rules, we present a method for true optimization of the response of dynamically excited, semiactively controlled structures subjected to constraints imposed by the dynamics of a particular semiactive device. The optimization procedure involves solving Euler–Lagrange equations. The closed-loop dynamics of structures with semiactive control systems are nonlinear due to the parametric nature of the control actions. These nonlinearities preclude an analytical evaluation of Laplace transforms. In this paper, frequency response functions for semiactively controlled structural systems are compiled from the computed time history responses to sinusoidal and pulse-like base excitations. For control devices with no saturation forces, the closed-loop frequency response functions are independent of the excitation amplitude. We make use of this homogeneity of the solution of semiactive control systems and present results in dimensionless form.     

Optimal Control of Earthquake Response Using Semiactive Isolation

The responses of two, low-rise, 2-degree-of-freedom base isolated structures with different isolation periods to a set of near-field earthquake ground motions are investigated under passive linear and nonlinear viscous damping, two pseudoskyhook semiactive control methods, and optimal semiactive control. The structures are isolated with a low damping elastic isolation system in parallel with a controllable damper. The optimal semiactive control strategy minimizes an integral norm of superstructure absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied and is determined through a numerical solution to the Euler–Lagrange equations. The optimal closed-loop performance is evaluated for a controllable damper and is compared to passive viscous damping and causal pseudoskyhook control rules. Results obtained from eight different earthquake records illustrate the type of ground motions and structures for which semiactive damping is most promising.     

Instantaneous Optimal Wilson-Θ Control Method

A control algorithm based on the instantaneous optimal control method, is presented for on-line control of the civil engineering structures subjected to earthquake excitations. This algorithm employs the unconditionally stable Wilson-θ method to discretize the continuous second-order form of the dynamical equation of motion, and uses a new performance index having an acceleration term as well as velocity and displacement terms. This method is named the instantaneous optimal Wilson-θ method. An eight-story shear-type building frame using one active mass damper/driver mechanism installed on the roof is used to demonstrate the efficiency of the proposed control algorithm in comparison to the other optimal control methods. Behavior of different combinations of weighting matrices related to displacement, velocity, and acceleration response vectors of the entire building are examined, and compared with the complete feedback control of the response vectors.     

Wavelet-Hybrid Feedback Linear Mean Squared Algorithm for Robust Control of Cable-Stayed Bridges

Cable-stayed bridges are flexible structures, and control of their vibrations is an important consideration and a challenging problem. In this paper, the wavelet-hybrid feedback least mean squared algorithm recently developed by the writers is used for vibration control of cable-stayed bridges under various seismic excitations. The effectiveness of the algorithm is investigated through numerical simulation using a benchmark control problem created based on an actual semifan-type cable-stayed bridge design. The performance of the algorithm is compared with that of a sample linear quadratic Gaussian (LQG) controller using three different earthquake records: the El Centro (California, 1940), Mexico City (Mexico, 1985), and Gebze (Turkey, 1999) earthquakes. Simulation results demonstrate that the new algorithm is consistently more effective than the sample LQG controller for all three earthquake records. Additional numerical simulations are performed to evaluate the sensitivity of the new control algorithm. It is concluded that the algorithm is robust against the uncertainties existing in modeling structures.     

Seismic Energy Dissipation of Inelastic Structures with Tuned Mass Dampers

The energy transfer process of using a tuned mass damper (TMD) in improving the ability of inelastic structures to dissipate earthquake input energy is investigated. Inelastic structural behavior is modeled by using the force analogy method, which is the backbone of analytically characterizing the plastic energy dissipation in the structure. Numerical simulations are performed to study the energy responses of structures with and without TMD installed. The effectiveness of TMD in reducing energy responses is also studied by using plastic energy spectra for various structural yielding levels. Results show that the use of TMD enhances the ability of the structures to store larger amounts of energy inside the TMD that will be released at a later time in the form of damping energy when the response is not at a critical state, thereby increasing the damping energy dissipation while reducing the plastic energy dissipation. This reduction of plastic energy dissipation relates directly to the reduction of damage in the structure, and TMD is therefore concluded to be quite effective in protecting structures from suffering major damage during an earthquake. However, storing energy in the TMD is restricted if the structure becomes plastic at a small displacement level. In this case, the effectiveness of TMD diminishes, and the structural response becomes practically the same as those without TMD installed.     

Model Predictive Control of Wind-Excited Building: Benchmark Study

In this paper, a “third generation” benchmark problem that focuses on the control of wind excited response of a tall building, using the Model Predictive Control (MPC) scheme, is presented. A 76 story, 306 m tall concrete office tower proposed for the city of Melbourne, Australia, is being used to demonstrate the effectiveness of MPC. The MPC scheme is based on an explicit use of a prediction model of the system response to obtain the control actions by minimizing an objective function. Optimization objectives in MPC include minimization of the difference between the predicted and desired response trajectories, and the control effort subjected to prescribed constraints. By incorporating input/output hard constraints, the MPC scheme provides an optimal control force that satisfies the prescribed constraints.     

Active Control of Structures in Nonlinear Response

The need to account for geometric and material nonlinearities in the active control of highly flexible, large, space structures is emphasized herein. The performance index of the control problem is minimized subject to equations of state and costate using a variable metric algorithm. Unlike the conventional techniques for active control, the present algorithm is able to exploit the sparsity and symmetry of the mass and stiffness matrices of the finite element models of structures. The algorithm thus has the potential of being able to control moderately large‐scale, finite element models of highly flexible, large, space structures in a cost‐effective manner. The proposed algorithm is validated in suppressing the nonlinear vibrations of an impulsively loaded, highly flexible beam, and the need for inclusion of nonlinearities is demonstrated.     

Effect of Asynchronous Earthquake Motion on Complex Bridges. I: Methodology and Input Motion

Based on observed damage patterns from previous earthquakes and a rich history of analytical studies, asynchronous input motion has been identified as a major source of unfavorable response for long-span structures, such as bridges. This study is aimed at quantifying the effect of geometric incoherence and wave arrival delay on complex straight and curved bridges using state-of-the-art methodologies and tools. Using fully parametrized computer codes combining expert geotechnical and earthquake structural engineering knowledge, suites of asynchronous accelerograms are produced for use in inelastic dynamic analysis of the bridge model. Two multi-degree-of-freedom analytical models are analyzed using 2,000 unique synthetic accelerograms with results showing significant response amplification due to asynchronous input motion, demonstrating the importance of considering asynchronous seismic input in complex, irregular bridge design. The paper, Part 1 of a two-paper investigation, presents the development of the input motion sets and the modeling and analysis approach employed, concluding with sample results. Detailed results and implications on seismic assessment are presented in the companion paper: Effect of Asynchronous Motion on Complex Bridges. Part II: Results and Implications on Assessment.     

Multiobjective Optimal Fuzzy Logic Controller Driven Active and Hybrid Control Systems for Seismically Excited Nonlinear Buildings

The third generation benchmark control problem for seismically excited nonlinear buildings is an effort to evaluate the developed control strategies in order to apply them in field applications. As the fuzzy logic control systems have been applied effectively in various fields, including vibration control of structures, a multiobjective optimal fuzzy logic control system has been proposed in this paper. Two types of control devices, namely, active and hybrid, driven by a fuzzy logic controller (FLC) have been considered in the present study. Nondimensionalized peak interstory drift ratio and peak floor acceleration have been used as the two objective functions for the multiobjective optimal design problem. A two-branch tournament genetic algorithm has been used to find a set of Pareto-optimal solutions, as the optimization problem is not necessarily continuous or convex. Performance of the FLC driven active and hybrid control systems have been evaluated for all three third generation benchmark problems for seismically excited nonlinear buildings (3-, 9- and 20-story). Acceleration and velocity information of different floors have been used as feedback to the FLC. This approach provides a set of Pareto optimal designs, from which a controller design can be selected for the required performance. The FLC driven active control system performs better than the sample controller given in the benchmark problem. Though the number of sensors and control devices are far less, the performance of the hybrid is close to the active control system.     

Space Structures: Issues in Dynamics and Control

A selective technical overview is presented on the vibration and control of large space structures, the analysis, design, and construction of which will require major technical contributions from the civil∕structural, mechanical, and extended engineering communities. The immediacy of the U.S. space station makes the particular emphasis placed on large space structures and their control appropriate. The space station is but one part of the space program, and includes the lunar base, which the space station is to service. This paper attempts to summarize some of the key technical issues and hence provide a starting point for further involvement. The first half of this paper provides an introduction and overview of large space structures and their dynamics; the latter half discusses structural control, including control‐system design and nonlinearities. A crucial aspect of the large space structures problem is that dynamics and control must be considered simultaneously; the problems cannot be addressed individually and coupled as an afterthought.     

Numerical Evaluation of Adaptive Base-Isolated Structures Subjected to Earthquake Ground Motions

In this study, the seismic response of a scale-model, base-isolated, multistory structure is numerically investigated. At the isolation level, the structure is equipped with isolation bearings combined with adaptive (semiactive) fluid dampers. The behavior of the dampers is controlled according to an H∞ optimal feedback control algorithm which utilizes a fuzzy logic approach to establish weighting functions. Numerical simulations are performed to evaluate the dynamic response of the isolated test structure when different damping mechanisms (passive, semiactive, or active) are incorporated within the isolation system. The numerical simulations indicate that, in comparison to an isolation system with high passive damping, an isolation system with semiactive damping can be effective in simultaneously controlling the response of the structure while limiting the isolation system response.     

Application of Centralized and Decentralized Control to Building Structure: Analytical Study

This paper presents both the centralized and decentralized control strategies by using semiactive control devices (magnetorheological dampers) for building structural control under seismic excitations. The control algorithms used in this study only focus on the linear quadratic Gaussian control theory with acceleration measurements as feedback signals. In the centralized control, the control force is generated directly from the measurement in each degree of freedom, while in the decentralized control, only measurement from local subsystems is used to calculate the control force with respect to the locations of control devices and measurements. Four groups of centralized/decentralized control algorithms are studied: fully centralized control, fully decentralized control, half-centralized control, and partially decentralized control. The ASCE 20-story benchmark building is selected as an example to evaluate the control performance among different control strategies.     

Seismic Analysis of Structures with Viscoelastic Dampers

Viscoelastic dampers are being used in structures to mitigate dynamic effects. The models of varying complexities from simple maxwell element to differential models with fractional and complex order derivatives have been used to represent their frequency-dependent force deformation characteristics. More complex models are able to capture the frequency dependence of the material properties better, but are difficult to use in analyses. However, the classical models consisting of assemblies of Kelvin and/or Maxwell elements with an adequate number of parameters can be formed to capture the frequency dependence as accurately as the more sophisticated fractional derivative models can do. The main advantage in adopting these classical models is a simpler and smaller system of equations, which can be conveniently analyzed for nonlinear and linear systems. In this study, the two classical mechanical models consisting of Kelvin chains and Maxwell ladder are used. It is shown that these mechanical models are as effective as the fractional derivative model in capturing the effect of the frequency dependence of the material properties in response calculations and are more convenient to use in dynamic response analyses.     

A Study of Two Stochastic Search Methods for Structural Control

Many engineering tasks involve the search for good solutions among many possibilities. In most cases, tasks are too complex to be modeled completely and their solution spaces often contain local minima. Therefore, classical optimization techniques cannot, in general, be applied effectively. This paper studies two stochastic search methods, one well-established (simulated annealing) and one recently developed (probabilistic global search Lausanne), applied to structural shape control. Search results are applied to control the quasistatic displacement of a tensegrity structure with multiple objectives and interdependent actuator effects. The best method depends on the accuracy related to requirements defined by the objective function and the maximum number of evaluations that are allowed.     

Improved Decentralized Method for Control of Building Structures under Seismic Excitation

A decentralized control method with improved robustness and design flexibility is proposed for reducing vibrations of seismically excited building structures. In a previous study, a control scheme was developed for multistory building models using nonlinear, decentralized control theory. This control method has now been improved in this study in that less information about material properties and geometrical parameters of the building is needed and the selection of control design parameters is more flexible. The nonlinear behavior of the proposed control system is studied and its stability property is proven mathematically. To evaluate the effectiveness and robustness of the proposed method, three illustrative structural models, i.e., an eight-story elastic shear beam model, a two-story nonlinear elastic shear beam model, and a 20-story elastic benchmark model are studied. The 1940 El Centro and the 1995 Kobe earthquakes are used in these examples. The performance of the current control design, as applied to these examples, has shown to be more effective in reducing structural responses and improving robustness.