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.     

Optimal H∞ Output Feedback Control Systems with Time Delay

An H∞ direct output feedback control algorithm through minimizing the entropy, a performance index measuring the tradeoff between H∞ optimality and H2 optimality, is developed in this paper to reduce the earthquake response of structures. To achieve optimal control performance and assure control system stability, the strategy to select both control parameters γ and α is extensively investigated considering the control force execution time delay. It is found that a lower bound of γ and an upper bound of α exist. The selection beyond these values will cause the control system instability. For a damped structure, analytical expressions of direct output feedback gains, controlled frequencies and damping ratios are derived. It can be proved that the conventional LQR control is a special case of the developed H∞ control. In real active control, control force execution time delay cannot be avoided. This paper gives explicit formulas of maximum allowable delay time and critical control parameters for the design of a stable control system. Some solutions are also proposed to lengthen maximum allowable delay times.     

Optimal Control Method for Seismically Excited Building Structures with Time-delay in Control

Optimal control method for seismic-excited linear structures with time delay in control is investigated in this paper. Using zero-order holder, the continuous time differential equation with time delay can be transformed into a standard discrete time form that contains no time delay in terms of two cases that the time delay is integer and noninteger times of sampling period, respectively. The continuous time performance index is used in the design of the optimal controller and it is also transformed into discrete form. Then, the optimal controller can be designed according to the classical discrete LQR method. The controller obtained contains not only current step of state feedback but also a linear combination of some former steps of control. Because the optimal controller is obtained directly from the time-delay differential equation, it is prone to guarantee the stability of the controlled structures. Furthermore, this control method is available for case of large time delay. The performance of the control method proposed and system stability are both demonstrated by numerical simulation results. Simulation results demonstrate that the control method proposed in this paper is a viable and attractive control strategy for application to seismically excited linear structures.     

Identification of Parametric Variations of Structures Based on Least Squares Estimation and Adaptive Tracking Technique

An important objective of health monitoring systems for civil infrastructures is to identify the state of the structure and to detect the damage when it occurs. System identification and damage detection, based on measured vibration data, have received considerable attention recently. Frequently, the damage of a structure may be reflected by a change of some parameters in structural elements, such as a degradation of the stiffness. Hence it is important to develop data analysis techniques that are capable of detecting the parametric changes of structural elements during a severe event, such as the earthquake. In this paper, we propose a new adaptive tracking technique, based on the least-squares estimation approach, to identify the time-varying structural parameters. In particular, the new technique proposed is capable of tracking the abrupt changes of system parameters from which the event and the severity of the structural damage may be detected. The proposed technique is applied to linear structures, including the Phase I ASCE structural health monitoring benchmark building, and a nonlinear elastic structure to demonstrate its performance and advantages. Simulation results demonstrate that the proposed technique is capable of tracking the parametric change of structures due to damages.     

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.     

Multiscale Wavelet-LQR Controller for Linear Time Varying Systems

This paper proposes a multiresolution based wavelet controller for the control of linear time varying systems consisting of a time invariant component and a component with zero mean slowly time varying parameters. The real time discrete wavelet transform controller is based on a time interval from the initial until the current time and is updated at regular time steps. By casting a modified optimal control problem in a linear quadratic regulator (LQR) form constrained to a band of frequency in the wavelet domain, frequency band dependent control gain matrices are obtained. The weighting matrices are varied for different bands of frequencies depending on the emphasis to be placed on the response energy or the control effort in minimizing the cost functional, for the particular band of frequency leading to frequency dependent gains. The frequency dependent control gain matrices of the developed controller are applied to multiresolution analysis (MRA) based filtered time signals obtained until the current time. The use of MRA ensures perfect decomposition to obtain filtered time signals over the finite interval considered, with a fast numerical implementation for control application. The proposed controller developed using the Daubechies wavelet is shown to work effectively for the control of free and forced vibration (both under harmonic and random excitations) responses of linear time varying single-degree-of-freedom and multidegree-of-freedom systems. Even for the cases where the conventional LQR or addition of viscous damping fails to control the vibration response, the proposed controller effectively suppresses the instabilities in the linear time varying systems.     

Optimal Neurocontroller for Nonlinear Benchmark Structure

A neurocontrol method is applied to the nonlinear benchmark control problem. A neurocontroller is trained based on a reduced-order linear design model, then it is used to control a nonlinear evaluation model. In training the controller, a sensitivity evaluation scheme is used and weights are updated by minimizing a cost function. Absolute accelerations directly measured from sensors are used as the feedback signals for the controller. Not only the current step acceleration, but delay signals of sensor readings, are used to enhance the training capability. Numerical examples show that the controlled responses are considerably reduced compared with the uncontrolled case. In conclusion, the possibility of the proposed control algorithm as a candidate for the controller of nonlinear building is shown.     

Neural Networks for Decentralized Control of Cable-Stayed Bridge

The system identification and vibration control of a cable-stayed bridge are considered difficult to achieve due to the bridge’s structural complexity and system uncertainties. In this paper, based on the concept of decentralized information structures, a decentralized, nonparametric identification and control algorithm with neural networks is proposed for the purpose of suppressing the vibration of a documented six-cable-stayed bridge model induced by earthquake excitations. The control strategy proposed here uses the stay cables as active tendons to provide control forces through appropriate actuators. Each individual actuator is controlled by a decentralized neurocontroller that only uses local information. The feature of decentralized control simplifies the implementation of the control algorithms and makes decentralized control easy to practice and cost effective. The effectiveness of the decentralized identification and control algorithm based on neural networks is evaluated through numerical simulations. And the adaptability of the decentralized neurocontrollers for different kinds of earthquake excitations and for a damaged cable-stayed bridge model is demonstrated via numerical simulations.     

Active Pulse Structural Control Using Artificial Neural Networks

Increasing interest has focused on applying active control systems to civil engineering structures subjected to dynamic loading. This study presents an active pulse control algorithm, termed the adaptive neural structural active pulse (ANSAP) controller, to control civil engineering structures under dynamic loading. The ANSAP controller minimizes structural cumulative responses during earthquakes by applying active pulse control forces. The effect of pulses is assumed to be delayed until just before the next sampling time so that the control force can be calculated in time and applied; the newly developed control strategy circumvents the effect of time delay due to the computation time. The ANSAP controller also circumvents the difficulty of obtaining system parameters of a real structure for the algorithm for active structural control. Illustrative examples reveal significant reductions in cumulative structural responses, which demonstrates the feasibility of using the adaptive artificial network for controlling civil engineering structures under dynamic loading.     

Adaptive Feedback–Feedforward Control of Building Structures

This paper proposes a combined feedback–feedforward control algorithm for reducing structural response of buildings to seismic excitations. The controller contains both feedback and feedforward components. The feedback component is assumed to be the same as that found from traditional linear quadratic regulator design. The feedforward component is obtained by estimating the external excitation as a series of step functions at each time increment. This feedforward gain varies with the duration of the step function used for estimation and converges as the time duration increases. Thus, a finite number of precalculated gains can be used to represent the potential feedforward gain profile. At any instant in time, the excitation is measured and by using the past measurements, the most effective feedforward gain for the recorded excitation values can be selected from the set of precalculated gains. This value is used as the feedforward gain for the current time step. Numerical examples are presented to show the effectiveness of this adaptive control scheme. The effects of varying the control objectives, the updating time for the feedforward gain, and the number and location of actuators are studied.     

Semiactive Control of the 20-Story Benchmark Building with Piezoelectric Friction Dampers

A new control algorithm is proposed in this paper to control the responses of a seismically excited, nonlinear 20-story building with piezoelectric friction dampers. The passive friction damping mechanism is used for low-amplitude vibration while the active counterpart takes over for high-amplitude vibration. Both the stick and sliding phases of dampers are taken into account. To effectively mitigate the peak story drift and floor acceleration of the 20-story building, multiple dampers are placed on the 20-story building based on a sequential procedure developed for optimal performance of the dampers. Extensive simulations indicate that the proposed semiactive dampers can effectively reduce the seismic responses of the uncontrolled building with substantially less external power than its associated active dampers, for instance, 67% less under the 1940 El Centro earthquake when the passive friction force is equal to 10% of the damper capacity.     

Time‐Delay Effect on Dynamic Response of Actively Controlled Structures

The effect of time‐delayed control signals on the stability and the dynamic response of actively controlled structures is investigated. The necessary optimal control law that takes the effect into account is derived using the variational approach. With the help of an example of a moment‐controlled simply supported beam subjected to a concentrated load, it is shown how time delay cannot only degrade the system response, but can also destabilize actively controlled structures. A gain‐compensating technique is proposed, and its use is illustrated with the help of the example. The results indicate that the effect of smaller values of time delay can be compensated for by applying additional energy, but the effects of higher values may not be compensated for, because of the prohibitively larger amounts of energy required in those situations.     

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.     

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.     

Cerebellar Model Articulation Controller (CMAC) for Suppression of Structural Vibration

Applicability of the cerebellar model articulation controller (CMAC) to structural control is explored, and a training algorithm for the suppression of structural vibration is proposed. The training epochs and controlled responses by both the CMAC and the well-known multilayer neural network are compared in the numerical example. The dynamics of the hydraulic actuator and the time delay of the controller are considered in the simulation. It is concluded that the CMAC is a promising candidate for a structural controller.     

Bayesian State Estimation Method for Nonlinear Systems and Its Application to Recorded Seismic Response

The focus of this paper is to demonstrate the application of a recently developed Bayesian state estimation method to the recorded seismic response of a building and to discuss the issue of model selection. The method, known as the particle filter, is based on stochastic simulation. Unlike the well-known extended Kalman filter, it is applicable to highly nonlinear systems with non-Gaussian uncertainties. The particle filter is applied to strong motion data recorded in the 1994 Northridge earthquake in a seven-story hotel whose structural system consists of nonductile reinforced-concrete moment frames, two of which were severely damaged during the earthquake. We address the issue of model selection. Two identification models are proposed: a time-varying linear model and a simplified time-varying nonlinear degradation model. The latter is derived from a nonlinear finite-element model of the building previously developed at Caltech. For the former model, the resulting performance is poor since the parameters need to vary significantly with time in order to capture the structural degradation of the building during the earthquake. The latter model performs better because it is able to characterize this degradation to a certain extent even with its parameters fixed. For this case study, the particle filter provides consistent state and parameter estimates, in contrast to the extended Kalman filter, which provides inconsistent estimates. It is concluded that for a state estimation procedure to be successful, at least two factors are essential: an appropriate estimation algorithm and a suitable identification model.     

Benchmark Control Problems for Seismically Excited Nonlinear Buildings

This paper presents the problem definition and guidelines of a set of benchmark control problems for seismically excited nonlinear buildings. Focusing on three typical steel structures, 3-, 9-, and 20-story buildings designed for the SAC project for the Los Angeles, California region, the goal of this study is to provide a clear basis to evaluate the efficacy of various structural control strategies. A nonlinear evaluation model has been developed that portrays the salient features of the structural system. Evaluation criteria and control constraints are presented for the design problems. The task of each participant in this benchmark study is to define (including sensors and control algorithms), evaluate, and report on their proposed control strategies. These strategies may be either passive, active, semiactive, or a combination thereof. The benchmark control problems will then facilitate direct comparison of the relative merits of the various control strategies. To illustrate some of the design challenges, a sample control strategy employing active control with a linear quadratic Gaussian control algorithm is applied to the 20-story building.     

Semianalytical Solution of Wave-Controlled Impact on Composite Laminates

Based on a structural model for wave-controlled impact, a modified Hertzian contact law was used to investigate the impact responses of composite laminates. The original nonlinear governing equation was transformed into two linear equations using asymptotic expansion. Closed-form solution can be derived for the first linear homogeneous equation, which is the equation of motion for single degree of freedom system with viscous damping. The second linear nonhomogeneous equation was solved numerically. The overall impact responses for wave-controlled impacts can be obtained semianalytically and agree well with the numerical solutions of nonlinear governing equations. The proposed methodology is useful for providing guidance to numerical simulation of impact on complex composite structures with contact laws fitting from experimental data.     

Potential for Structural Failure in the Seismic Near Field

This study analyzes the possibility that large distortions and distortion rates due to wave-propagation phenomena within structures were responsible for unexpected cracking at connections of steel-frame buildings in the seismic near-field region during the Northridge (1994) and Kobe (1995) earthquakes. Since such internal wave propagation is characteristic of a structure with a continuous distribution of mass, the problem is studied by numerically simulating the structural response for both discrete and continuous models of a 20-story building, using ground motion time histories from the Northridge earthquake. The time histories are chosen from the far-field and near-field regions of the earthquake to determine if wave-propagation effects within the structure are especially significant in the near field. A truncated modal analysis is also performed using only the first vibrational mode to see if significantly lower response levels result. It is found that the continuous model gives higher response levels—indicating that wave propagation may have been a factor—but the discrepancy is not limited to the near field. Strain rates are higher from the continuous model than from the discrete model and much higher than from the truncated modal analysis, but the magnitudes are too low to be a significant factor in the observed damage. The explanation for the connection cracking may simply be high-intensity ground motion in the near field.