Simple adaptive control to attenuate bridge’s seismic responses considering parametric variations

Abstract

Parametric variations occur throughout a bridge’s service life as a result of temperature variations, cracking, localized damage, and fatigue. Existing bridge’s parameters are difficult to estimate precisely and implemented control schemes may perform unsatisfactorily depending on how sensitive they are to parametric change. In this study, an adaptive control approach is developed utilizing the simple adaptive control algorithm and designed aiming to mitigate seismic responses of bridges considering realistic implementation. Adaptive control is a viable alternative to control bridge structures as it is able to calculate control gains that vary over time based on sensed responses, sustaining performance in face of parametric variations. The proposed approach allows the choice of a model reference of significantly low order and it does not require full-state feedback or the use of observers. The effectiveness and robustness of the control approach are investigated when controlling a seismically excited two-span highway bridge considering systematic parametric variations. Evaluation of relevant performance criteria indicates that the adaptive scheme is effective in reducing seismic responses and sustains well the overall performance when systematic parametric variations are introduced. The proposed approach requires a small and reasonable number of sensors to present effectiveness and performs satisfactorily even when in the presence of measurement noise.

Keywords

adaptive control bridge seismic control parametric variation semi-active control simple adaptive control

Introduction

Bridge structures typically have little structural redundancy and are prone to present undesired excessive responses induced by dynamic extreme events. Dynamic structural control is a modern alternative to attenuate dynamic responses caused by extreme events and ensure that dynamic responses remain below an acceptable value. However, parametric estimation of bridges is tied to a lot of uncertainties. Estimation of mass, stiffness, and damping, temperature variations, modeling assumptions, and simplifications are likely to lead to estimated parameters that are different from those of the existing bridge. In addition, changes that occur during the bridge service life as a result of changes in temperature, cracking, accumulation of snow, and damage all lead to parametric variations. As a result, the originally designed control strategy may present an unsatisfactory performance depending on how sensitive it is to parametric changes. It is important to point out that, given the size and nature of bridge structures, it is particularly difficult to estimate precisely mass, stiffness, and damping parameters. It is desired that the designed controller sustains performance in face of parametric variations to guarantee that the control scheme performs satisfactorily when an extreme event takes place. In light of this issue, adaptive control strategies may be a good alternative to control bridge structures, as they can calculate control gains that vary over time and are based on existing plant real-time sensed responses.

Adaptive control schemes for building structures have been implemented successfully in many studies. In Gattulli and Romeo (2000), an integrated procedure is proposed to identify and control a multi-degree-of-freedom (MDOF) structure. The effectiveness of the adaptive control algorithm sliding mode control and model reference adaptive control (MRAC) integrated to an online parameter identification procedure based on tracking errors are investigated. Numerical simulations are performed in a three-degree-of-freedom shear-building-type structure and the proposed method is successful in identifying the changes in parameters and control excessive vibrations. In the study developed by Schurter and Roschke (2001), a neuro-fuzzy strategy is implemented with acceleration feedback to control buildings with magnetorheological (MR) dampers. A single-degree-of-freedom (SDOF) and a shear building are subjected to different earthquake records and the performance of the semi-active scheme is compared to that of purely passive control. In Chu (2009), a real-time model reference adaptive identification technique is proposed in order to incorporate online system identification into the MRAC algorithm. The law used for parameter estimation is based on Lyapunov’s direct method, and the energy function is considered by assembling weighted response tracking error and parameter estimation error. A numerical simulation for an SDOF time-invariant system is performed and the system is subjected to two different sets of earthquake loads. The proposed control is shown to be effective, as the parameters of the system are successfully identified and excessive vibrations attenuated. In Bitaraf et al. (2010), adaptive control was implemented to control a seismically excited three-story building. The ability of the method to deal with different damage scenarios was assessed, and the damaged controlled structure performed in a very similar manner as the undamaged one. In Bitaraf et al. (2012), a semi-active adaptive scheme was implemented to control a three-story building, presenting satisfactory performance even when in the presence of noise and damage. Bitaraf and Hurlebaus (2013) implemented an adaptive scheme to successfully control a 20-story building taking into consideration non-linearities’ effects. In Al-Fahdawi et al. (2018), simple adaptive control (SAC) and MR dampers were used to mitigate excessive seismic response of multi-story adjacent buildings. The study considered stiffness and mass reduction in order to simulate damage scenarios. The results indicated the effectiveness of the adaptive control semi-active scheme in both attenuating seismic response and sustaining performance in face of parametric variation. Adaptive schemes were also applied successfully to control tall buildings (Hosseini and Taghikhany, 2018; Venanzi et al., 2017) and non-linear systems (Javanbakht, 2016; Ulrich and Sasiadek, 2014).

Adaptive schemes to control bridges have not been deeply studied, especially considering realistic implementation conditions and systematic parametric variations. In Ningsu (1999), a decentralized model reference adaptive controller was designed to reduce excessive transverse vibration of a cable-stayed beam under seismic loading. The vibrations were significantly reduced by the proposed controller when compared to the uncontrolled case. The SAC strategy was applied to control a three-span continuous bridge subjected to earthquake excitation in Soares et al. (2018). Stiffness reduction was introduced and the robustness of the controller was assessed. The adaptive scheme is shown to be effective in sustaining and reducing the response of the bridge. The aforementioned studies indicate that adaptive schemes are suitable for controlling bridges and are able to reduce vibrations successfully.

This study presents adaptive control as an alternative to control bridge structures in face of parametric variations. An adaptive control approach utilizing the SAC algorithm is developed aiming to mitigate seismic responses of bridges considering realistic implementation. SAC is an adaptive control technique based on the classical MRAC. The idea of SAC was first introduced by Sobel et al. (1982) and has been developed over a series of studies (Barkana, 1987, 2005, 2008, 2013, 2014, 2016a, 2016c; Barkana and Guez, 1990; Barkana and Kaufman, 1993). The idea behind SAC is to overcome limitations of the classic MRAC for multiple-input multiple-output (MIMO) systems, such as requiring full-state feedback or full-order observers, and instability caused by unmodeled dynamics. In Barkana (2014), a survey is presented containing the method’s latest developments. In this study, a review and stability proof under ideal conditions of the MRAC are presented. Provided that the system is stable and the full state vector is available, the MRAC satisfies a strict positive realness condition and it is proven stable. However, it is pointed out that unmodeled dynamics that occur when the actual structural system is of higher order than the model reference have the potential to lead to instability in the presence of disturbances or noise. SAC can be implemented with a significantly reduced order model reference when compared to the actual structural system, and it is applicable to systems that are prone to the instability present. Moreover, SAC is proven to guarantee perfect tracking asymptotically and it successfully avoids the need of estimators.

The scheme is promising to control bridge structures given that it allows the choice of a model reference of significantly low order and it does not require full-state feedback or the use of observers. In addition, the adaptive technique is able to deal well with significant disturbances, and therefore it is suitable to be utilized to mitigate seismic responses. Adaptive control is presented as an alternative to control bridge structures subjected to seismic excitation; as bridge structures are significant in size, there are many uncertainties involved in the estimation of bridges’ structural parameters, and bridges are susceptible to many external elements that may change their structural parameters significantly.

The control approach is designed and implemented for a seismically excited two-span highway bridge, and its effectiveness is assessed considering systematic parametric variations. The adopted reference tracking direct adaptive control algorithm allows a model reference with a significantly lower order than the actual structure, it is proven to guarantee perfect tracking, and it does not require full-state feedback or the use of observers. The strategy is designed and implemented to provide control command for MR dampers installed at the bridge ends. The scheme is assessed by subjecting the bridge to a set of 11 earthquakes, while stiffness and mass parameters are varied systematically. The performance of the adaptive semi-active control scheme is compared to those of non-adaptive passive control and semi-active optimal control, taking into account the effects of noise and device dynamics.

Simple adaptive control (SAC)

SAC is a direct model reference adaptive strategy. The method does not explicitly calculate plant parameters in order to obtain the control gains necessary for tracking a desired behavior. SAC is based on the classical MRAC and it overcomes many of the drawbacks of the classical method when it comes to the implementation of MIMO systems. SAC gives the possibility of adopting a significantly reduced order model when compared to the plant and is applicable to plants that are possibly unstable. SAC is proven to guarantee perfect tracking asymptotically and does not require full-state feedback or the use of identifiers or observers. The control algorithm is promising in overcoming parametric changes, disturbances, and noise (Barkana, 2016c). Figure 1 shows the block diagram of the SAC algorithm. SAC’s control law is given by equation (1), where r(t) is a matrix composed of e_y, x_m, and u_m, the error between the output of the model reference and the plant, the model states, and the model reference input, respectively. r(t) is given by equation (3). The time-varying control gain matrix K(t) is given by the differential equations (4) to (6)

u_{p} (t) = K (t) r (t)

(1)

K (t) = [K_{e} (t), K_{x} (t), K_{u} (t)]

(2)

r^{T} (t) = [y_{m} - y_{p} {x_{m}}^{T} (t) {u_{m}}^{T} (t)]

(3)

{\overset{\cdot}{K}}_{e} = e_{y} (t) e_{y} {(t)}^{T} Γ_{e}

(4)

{\overset{\cdot}{K}}_{x} = e_{y} (t) x_{m} {(t)}^{T} Γ_{x}

(5)

{\overset{\cdot}{K}}_{u} = e_{y} (t) u_{m} {(t)}^{T} Γ_{u}

(6)

Figure 1.

SAC’s block diagram.

SAC-guaranteed stability depends on the plant transfer function being almost strictly positive real (ASPR), and most real-world systems cannot be guaranteed to satisfy this condition. In the past, a sigma term had been used to guarantee stability under disturbances (Ioannou and Kokotovic, 1984a, 1984b); however, it has been observed that this term could eliminate perfect tracking and lead to chaotic-like phenomena. A parallel feedforward (PFC) term was introduced (Barkana, 2016b) to guarantee perfect tracking and robustness under disturbances and non-ideal scenarios, eliminating the need for adding the sigma term. The PFC configuration added to the plant guarantees ASPR conditions.

Linear quadratic Gaussian control (LQG)

The combination of a linear quadratic regulator (LQR) and a linear quadratic estimator (LQE) leads to a linear quadratic Gaussian (LQG) controller. The LQR is an optimal control strategy that pursues a suitable control solution while achieving the minimum cost (Burl, 1999). A regulator control problem seeks to optimally return a system initially displaced from the equilibrium position back to the equilibrium position. This is achieved by minimizing a performance index that can be defined as a measure of the quality of the system behavior. The LQR control law gives an optimal control solution for a linear system by minimizing the quadratic performance index J given by equation (7), where x(t) is the system state vector. The matrices Q and R are the weighting matrices, by changing the values of which the controller can be tuned. A very small R, for example, means fast convergence and high control efforts. Q is a symmetric semi-definite matrix and R is a symmetric and positive definite matrix

J = \int_{0}^{t_{f}} [x {(t)}^{T} Q x (t) + u {(t)}^{T} R u (t)] dt

(7)

The control law for the strategy is given by equation (8) and the gain K is obtained by equation (9). P is the unique solution of the non-linear matrix Riccati equation (10). LQR stability is easily guaranteed if all the states of the system are available for feedback (full feedback), and the system is well defined.

u (t) = - K x (t)

(8)

K = R^{- 1} B^{T} P

(9)

A^{T} P + PA - BP R^{- 1} B^{T} P + Q = 0

(10)

If it is not possible to measure all states, it is necessary to use an observer to reconstruct all the states from the measurable output. The stability in this case is still guaranteed, but the margins become smaller. One way of estimating the states is by implementation of the LQE (Chernousko, 1993). The LQE estimates the state in the presence of Gaussian noise in the output measurements. Given a system with output u, white process noise w, and white measurement noise v that are zero-mean stochastic Gaussian processes uncorrelated in time and with each other

\overset{\cdot}{x} = Ax + Bu + Gw

(11)

y = Cx + Du + Hw + v

(12)

Consider that they have the following covariance

E (w w^{T}) = W

(13)

E (v v^{T}) = V

(14)

LQE can be designed as given by equation (15), where $\hat{x}$ is the state estimate, L is the observer gain matrix given by equation (16), and P is the solution of the algebraic Riccati equation (17)

\overset{\cdot}{\hat{x}} = A \hat{x} + Bu + L (y - C \hat{x})

(15)

L = P C^{T} V^{- 1}

(16)

AP + P A^{T} - P C^{T} V^{- 1} P + W = 0

(17)

MR dampers

The MR damper is a semi-active controllable damper that provides dependable functionality with considerably low power requirements. It has the advantage of becoming passive dampers in case there is any malfunction, which guarantees a basic level of functionality (Spencer et al., 1997). MR dampers consist of dampers filled with an MR fluid. This fluid changes its viscosity properties when in face of an electromagnetic field. This feature turns the MR damper in an adjustable semi-active control device where the generated forces are compatible with the controller output. The major advantages of using MR dampers to control civil structures are as follows: the large yield stress it is able to achieve which impacts the device size and dynamic range; being able to be sourced by low-voltage batteries; and being less susceptible to dielectric breakdown, contamination, and extreme temperatures (Spencer and Sain, 1997). According to Spencer and Nagarajaiah (2003), these devices have the advantage of being mechanically simple, given that they contain no moving parts other than the piston. These dampers have been successfully implemented to control civil structures in a series of studies (Barroso et al., 2002; Bharti et al., 2010; Chen et al., 2004; Erkus et al., 2002; Jung, 2004; Jung et al., 2003).

The dynamic behavior of the device is obtained through the use of the simple Bouc–Wen model (Spencer et al., 1997) as shown in Figure 2. Its hysteretic behavior is represented through the Bouc–Wen model (Wen, 1976). The damper force is obtained by equation (18) where $\overset{\cdot}{x}$ is the system velocity. C₀ is obtained by equation (19) and α is obtained by equation (20). C_0a and C_0b are constants, and the input voltage u can be obtained by the inverse model described by equation (21), where u_c is the controller input, and equation (22), where v is the output voltage. The evolutionary variable z describes the hysteretic behavior and is obtained by equation (23), where γ, β, A, and n are constants. For this study, 1000-kN capacity MR dampers are used and the parameters are available in Yoshida and Dyke (2004)

f = C_{0} \overset{\cdot}{x} + α z

(18)

C_{0} = C_{0 a} + C_{0 b} u

(19)

α = α_{a} + α_{b} u

(20)

u = \frac{f - C_{0 a} \overset{\cdot}{x} - α_{a} z}{C_{0 b} \overset{\cdot}{x} + α_{b} z}

(21)

v = u + \frac{\overset{\cdot}{u}}{η}

(22)

\overset{\cdot}{z} = - γ | \overset{\cdot}{x} | z {| z |}^{(n - 1)} - β \overset{\cdot}{x} {| z |}^{n} + A \overset{\cdot}{x}

(23)

Figure 2.

Simple Bouc–Wen Model Schematics.

Numerical evaluation: parametric study

For the numerical evaluation and parametric study, a highway bridge is considered as an example and subjected to a set of different earthquakes. The structural characterization is based on the benchmark problem for seismically excited highway bridges (Agrawal et al., 2009). The bridge consists of a continuous two-span prestressed concrete box girder bridge highway overcrossing located in Southern California. Each span is 58.5 m long and the bridge spans a four-lane highway. The abutments are skewed at 30° and the deck width is 12.95 m, as shown in Figure 3. The cross section of the deck consists of three cells and is supported by a 31.4-m-long and 6.9-m-high prestressed outrigger supported by columns approximately 6.9 m high. The system rests on two pile groups, each consisting of 49 driven concrete friction piles. The cross section of the bridge along the transverse beam and a plan view of the pile group are shown in Figure 4. The bridge superstructure is represented by three-dimensional (3D) beam elements and rigid links are used to model the abutments and deck-ends. The effects of soil–structure interaction at the end abutments/approach embankments are included by frequency-independent springs and dashpots. The masses of the non-structural elements are included in the model and their stiffness contribution is neglected. All element mass matrices and initial elastic element stiffness matrices obtained through the finite element model are summed at nodal masses to assemble global stiffness and mass matrices within the MATLAB (MathWorks, Inc., 2017) environment. The inherent damping of the superstructure is assumed to be a function of the mass and initial elastic stiffness matrix of the superstructure. The Raleigh damping parameters are computed by assuming a 5% modal damping ratio in the first and second modes. Table 1 summarizes the modal properties of the bridge.

Figure 3.

Elevation and plan view of the highway bridge.

Figure 4.

Cross section along the center bent of highway bridge and configuration of pile groups.

Table 1.

Modal properties of the bridge.

Mode	Frequency (Hz)	Period (s)	Description
1	1.25	0.80	Torsional
2	1.28	0.78	Torsional and vertical
3	1.52	0.66	Vertical
4	1.68	0.59	Transverse
5	1.71	0.58	Vertical
6	3.24	0.31	Transverse

Material properties and member size estimation and thermal variations may lead to uniform changes in stiffness or mass. The following scenarios of parametric changes are considered: overall mass increase of 10% and 5%, overall mass reduction of 10% and 5%, overall stiffness increase of 25% and 10%, and overall stiffness reduction of 20% and 25%. In addition, it is considered a combination of an overall stiffness increase of 25% and an overall mass reduction of 10%, a combination of an overall stiffness increase of 25% and an overall mass increase of 10%, a combination of an overall stiffness reduction of 25% and an overall mass reduction of 10%, and a combination of an overall stiffness reduction of 25% and an overall mass increase of 10%.

Placement of devices and sensors

A total of 16 control devices, 8 at each bridge end, are placed between abutments and the bridge’s deck controlling the transverse and longitudinal directions. A total of 10 sensors measuring displacements are considered for the SAC scheme. LQR requires full-state feedback or states’ reconstruction through an estimator. For the reconstruction of the states to be of quality, it is necessary that the system is observable. Observability gives a measure of how well internal states of a system can be estimated by the measured outputs. A system is observable when the current state can be well determined from the output information. A total of 10 sensors measuring displacements, as adopted for the adaptive strategy, lead to a non-observable system which indicates that the estimator reconstruction of the states is not accurate. The LQR + LQE operating with 10 sensors is examined in this study for comparison purposes. Nevertheless, an observable system composed of LQR + LQE operating with a total of 41 sensors measuring displacements is also considered. Figure 5 shows the schematics of the sensors’ distribution. White noise is introduced into the measured outputs given that such measurements are likely to be imperfect and affected by noise.

Figure 5.

Sensors’ schematics: (a) SAC and LQR non-observable system—10 sensors measuring displacements and (b) LQR observable system—41 sensors measuring displacements.

The sensors’ quantity, distribution, and placement depend on the consideration of a number of factors. The number of sensors is obtained over a balance of what is economically and realistically feasible and what is enough for the particular control algorithm to work properly. For example, if full reconstruction of states is needed, observability is necessary. The placement and distribution of sensors depend mostly on the dynamic characteristics of the particular structural system and its structural configuration; key sensor locations are determined following the observation of modal contributions and mode shapes and the external load’s properties. In addition, access to the sensors must be taken into account, in order to facilitate installation and maintenance. For this study, the distribution of sensors shown in Figure 5(a) is found to be enough to measure the responses of key locations and for the adaptive control (SAC) to work properly, and it is positioned in locations of easy access for installation and maintenance purposes. The distribution is considered for the LQR + LQE scheme as a way to illustrate the necessity of observability for this particular control solution. Given that the sensors’ distribution from Figure 5(a) leads to a non-observable system and, as such, does not reconstruct states properly, an observable distribution is searched considering the aforementioned factors. A distribution found to provide an observable system is displayed in Figure 5(b) and also considered for the LQR + LQE solution. And, obviously, the distribution shown in Figure 5(a) is more appealing than the one in Figure 5(b), given that there are fewer sensors, which facilitates maintenance and installation and reduces cost. The sensor distributions considered in this study are not by any chance the only possible distributions but are found after careful consideration regarding the feasibility, the dynamic behavior of the bridge in question, and the particularities of the control schemes. The exploration of optimization rules is not performed in this particular study, but it is a valuable suggestion for future studies concerning control solutions for the mitigation of seismic responses of bridges.

From this point on, the semi-active adaptive control scheme proposed in this study composed of SAC with 10 sensors and MR dampers is referred to as SAC(a). The comparative semi-active optimal controller with 10 sensors and MR dampers is referred to as LQR(a) and the semi-active optimal controller with 41 sensors is referred to as LQR(b).

Earthquake suite

The earthquake suite is chosen as an attempt of covering a full range of different earthquakes’ characteristics. The set comprehends far- and near-field earthquakes, different values of moment magnitude, and different peak accelerations and velocities. Table 2 summarizes the characteristics of the earthquake suite and Figure 6 shows the acceleration response spectra for all the earthquakes considered in this study.

Table 2.

Earthquake suite characteristics.

Earthquake (year)	Record station	Magnitude (Mw)	Dist. to fault (km)	PGA (g)	PGV (m/s)	Duration (s)
Chi-Chi, Taiwan (1999)	TCU084	7.6	10.4	1.16	1.15	90
Duzce, Turkey (1999)	Bolu	7.1	17.6	0.73	0.56	56
Imperial Valley (1979)	El Centro Array #7	6.4	29.4	0.46	1.13	36.5
Kobe, Japan (1995)	Nishi-Akashi	6.9	11.1	0.51	0.37	41
Landers (1992)	Lucerne Valley	7.3	42	0.71	1.26	47
Loma Prieta (1989)	Los Gatos	7	6.1	0.56	0.95	39
Palm Springs (1986)	North Palm Springs	6	7.3	0.49	0.73	10
Northridge (1994)	Rinaldi	6.7	7.1	0.84	1.66	13
Petrolia (1992)	Cape Mendocino	7	3.8	1.50	1.25	17
San Fernando (1971)	Pacoima Dam	6.6	8.5	1.17	1.14	9
Superstition Hills (1987)	Parachute Test Site	6.6	7.2	0.45	0.99	15

PGA: peak ground acceleration; PGV: peak ground velocity.

Figure 6.

Acceleration response spectra for the earthquake suite.

Performance evaluation criteria

The performance evaluation criteria selected for this study are defined by equations (24) to (29) (Agrawal et al., 2009). J₃ gives peak displacement for the controlled structure normalized by the peak displacement of the uncontrolled structure; J₄ gives peak acceleration for the controlled structure normalized by the peak acceleration of the uncontrolled structure; J₁₁ gives normed displacement for the controlled structure normalized by the normed displacement of the uncontrolled structure; J₁₅ gives the peak control force normalized by the seismic weight of the superstructure; and J₂₀ is the number of sensors

J_{3} = max {max | \frac{y_{mi} (t)}{y_{0 m}^{max}} |}

(24)

J_{4} = max {max | \frac{{\overset{\cdot\cdot}{y}}_{mi} (t)}{{\overset{\cdot\cdot}{y}}_{0 m}^{max}} |}

(25)

‖ \cdot ‖ = \sqrt{\frac{1}{t_{f}} \int_{0}^{t_{f}} {(\cdot)}^{2}}

(26)

J_{11} = max {max \frac{‖ y_{mi} (t) ‖}{‖ y_{0 m}^{max} ‖}}

(27)

J_{15} = max {max | \frac{f_{i} (t)}{W} |}

(28)

J_{20} = number of sensors

(29)

Results and discussion

In this section, the results obtained in terms of the performance criteria established previously are presented and discussed. The responses displayed are for the midspan in the transverse direction (y-axis direction; see Figure 5). Table 3 presents J₁₅ and J₂₀ performance criteria values for all different control strategies. Figures 7 and 8 show the 3D plots of J₃ and J₁₁, respectively, for different mass and stiffness variations due to the Chi-Chi earthquake. Figure 9 shows a 3D plot of J₁₁ for different mass and stiffness variations for earthquake Duzce. Figure 10 shows a 3D plot of J₁₁ for different mass and stiffness variations for the Landers earthquake. Figure 11 shows the displacement time histories for different control strategies for the original design and the structure with reduced stiffness in 25% and an increase in mass of 10%, for the Petrolia earthquake. Figure 12 shows the maximum performance criteria J₃, J₄, and J₁₁ for different control strategies due to the Chi-Chi earthquake.

Table 3.

Performance criteria J₁₅ and J₂₀ values for the different control strategies.

Performance criteria	Passive-off	Passive-on	SAC(a)	LQR(a)	LQR(b)
J ₁₅	0.039	0.001	0.039	0.039	0.039
J ₂₀	–	–	10	10	41

SAC: simple adaptive control; LQR: linear quadratic regulator.

Figure 7.

J₃ for different scenarios of mass and stiffness changes for the Chi-Chi earthquake.

Figure 8.

J₁₁ for different scenarios of mass and stiffness changes for the Chi-Chi earthquake.

Figure 9.

J₁₁ for different scenarios of mass and stiffness changes for the Duzce earthquake.

Figure 10.

J₁₁ for different scenarios of mass and stiffness changes for the Landers earthquake.

Figure 11.

Displacement time histories for different control strategies for the Petrolia earthquake.

Figure 12.

Maximum performance criteria J₃, J₄, and J₁₁ for different control strategies for the Chi-Chi earthquake.

Table 4 shows the maximum for performance criteria J₃, J₄, and J₁₁ for each different control strategy, for all earthquakes considered in this analysis. Table 5 shows the standard deviation among the different parametric scenarios considered for the performance criteria J₃, J₄, and J₁₁ of each different control strategy.

Table 4.

Maximum performance criteria J₃, J₄, and J₁₁ among the different parametric variation scenarios.

Contr.	Chi-Chi	Duzce	Imp. Val.	Kobe	Land.	Loma Pr.	Palm Spr.	Nort.	Petr.	San Fern.	Sup. Hills
Max J₃
Unc.	1.308	1.066	1.634	1.348	1.289	1.523	1.093	1.263	1.165	1.129	1.217
P. On	1.261	1.064	1.452	2.013	1.510	1.372	1.033	1.163	1.032	1.879	1.138
P. Off	1.187	1.045	1.545	1.273	1.140	1.294	1.046	1.214	1.126	1.010	1.085
LQR(a)	1.336	1.052	1.482	1.377	1.348	1.643	1.114	1.162	1.093	1.123	1.270
LQR(b)	1.009	0.648	1.158	1.095	0.940	0.934	0.903	1.111	1.066	1.588	0.781
SAC(a)	0.853	0.715	1.132	1.070	0.988	1.172	1.051	0.959	0.966	1.186	0.806
Max J₄
Unc.	1.668	1.304	1.333	1.551	1.219	1.433	1.292	1.645	1.299	1.397	1.542
P. On	1.639	1.413	1.409	1.757	1.305	2.665	1.348	1.809	1.569	1.859	1.427
P. Off	1.596	1.277	1.280	1.489	1.071	1.297	1.180	1.598	1.336	1.287	1.400
LQR(a)	3.984	3.259	7.728	6.344	6.572	9.121	4.940	3.722	2.869	3.570	5.127
LQR(b)	2.190	1.796	2.945	2.692	2.592	3.573	1.701	1.920	1.912	1.877	2.166
SAC(a)	1.365	1.094	1.761	1.543	1.017	1.827	1.214	1.481	1.582	1.219	1.515
Max J₁₁
Unc.	1.702	1.106	1.338	1.227	1.574	1.275	1.009	1.361	1.531	1.551	1.150
P. On	0.900	1.002	1.032	1.277	1.404	1.051	0.592	1.522	1.325	1.725	0.839
P. Off	1.323	0.870	0.971	0.971	0.990	0.829	0.739	1.079	1.126	1.142	0.866
LQR(a)	1.856	1.317	1.780	1.391	2.110	1.651	1.088	1.147	1.602	1.761	1.282
LQR(b)	0.676	0.423	0.557	0.658	0.645	0.474	0.404	0.830	0.670	1.073	0.501
SAC(a)	0.892	0.461	0.742	0.664	0.747	0.607	0.641	0.804	0.669	0.980	0.549

LQR: linear quadratic regulator; SAC: simple adaptive control.

Table 5.

Performance criteria J₃, J₄, and J₁₁ standard deviation among the different parametric variation scenarios.

Contr.	Chi-Chi	Duzce	Imp. Val.	Kobe	Land.	Loma Pr.	Palm Spr.	Nort.	Petr.	San Fern.	Sup. Hills
Standard deviation—J₃
Unc.	0.084	0.059	0.282	0.095	0.148	0.239	0.052	0.161	0.104	0.074	0.155
P. On	0.241	0.088	0.219	0.366	0.263	0.148	0.014	0.162	0.038	0.322	0.142
P. Off	0.067	0.061	0.268	0.089	0.110	0.203	0.077	0.145	0.087	0.065	0.130
LQR(a)	0.089	0.062	0.272	0.098	0.167	0.216	0.074	0.118	0.054	0.067	0.159
LQR(b)	0.088	0.058	0.115	0.089	0.113	0.105	0.023	0.116	0.019	0.201	0.195
SAC(a)	0.078	0.031	0.102	0.085	0.082	0.190	0.110	0.109	0.024	0.172	0.084
Standard deviation—J₄
Unc.	0.373	0.182	0.312	0.265	0.223	0.228	0.240	0.282	0.215	0.199	0.317
P. On	0.185	0.269	0.238	0.242	0.211	0.597	0.333	0.308	0.367	0.217	0.291
P. Off	0.348	0.181	0.289	0.250	0.194	0.202	0.209	0.279	0.216	0.170	0.284
LQR(a)	3.001	2.388	5.797	4.553	4.881	6.858	3.761	2.738	2.234	2.705	5.139
LQR(b)	1.469	1.395	0.934	1.567	0.987	1.441	0.123	0.283	0.354	0.320	0.580
SAC(a)	0.161	0.119	0.293	0.099	0.110	0.200	0.121	0.209	0.276	0.097	0.202
Standard deviation—J₁₁
Unc.	0.252	0.131	0.274	0.082	0.176	0.217	0.126	0.223	0.197	0.222	0.084
P. On	0.119	0.167	0.135	0.198	0.088	0.163	0.058	0.216	0.187	0.160	0.134
P. Off	0.175	0.091	0.174	0.066	0.072	0.118	0.049	0.159	0.117	0.109	0.059
LQR(a)	0.281	0.149	0.369	0.067	0.287	0.275	0.110	0.169	0.183	0.263	0.103
LQR(b)	0.063	0.044	0.051	0.074	0.052	0.042	0.029	0.089	0.036	0.103	0.194
SAC(a)	0.120	0.033	0.069	0.042	0.019	0.029	0.042	0.124	0.016	0.085	0.029

LQR: linear quadratic regulator; SAC: simple adaptive control.

LQR(a) does not give an overall satisfactory performance, when considering criteria J₃, J₄, and J₁₁. Given the scheme non-observability, the states of the system are not well reconstructed. This is enough to affect the performance of the control solution significantly, which illustrates the importance of observability for the optimal strategy.

Observation of the performance criterion J₃ indicates that SAC(a) and LQR(b) exhibit the best performances in terms of reduction of peak displacements. Both strategies are able to reduce the overall peak displacement and sustain the performance well when in face of parametric variations. Observation of the performance criterion J₁₁ indicates that LQR(b) reduces the root mean square (RMS) displacements very well, providing the best performance. SAC(a) reduces also the RMS displacements well, providing the second best performance. The passive-off strategy is able to reduce overall peak and RMS displacements (J₃ and J₁₁ criteria). However, the reduction is not as significant as that obtained by SAC(a) and LQR(b), and the passive strategy is not able to sustain performance. Observation of the performance criteria J₃ and J₁₁ indicates that the passive-on strategy is able to reduce peak and RMS displacements for some earthquakes, while it worsens the peak displacements for the others. In addition, for some parametric change scenarios, the passive-on strategy increased significantly the midspan RMS and peak displacements. This response worsening is attributed to excessive stiffness introduced by the devices to the ends of the bridge. This phenomenon is observed when considerable placement constraints restrict the location of passive control devices. Given that for most bridges’ structural configurations the devices can only be installed in very specific locations, this issue is something to be regarded by designers.

Observation of the performance criterion J₄ shows that most control schemes increase the accelerations at some point, depending on the parametric variation scenario or the earthquake. For both passive strategies and SAC(a), there is no observation of significant worsening of acceleration responses, meaning that whenever there is observation of worsening of responses they are not substantial. For LQR(b), however, there is a considerable increase in acceleration responses, which can be a drawback of for the scheme.

Observation of the evaluation criterion J₁₅ indicates that peak control forces necessary for the passive-off strategy are the lowest among all strategies, and the other strategies all reach the same maximum control force. The numbers are an indication that the passive-off strategy reaches lower forces than the passive-on and the semi-active control schemes due to the force capacity limitation of the MR damper when it is turned off. All the other strategies are able to reach the peak force of the damper, which means that the device is designed to its full maximum capacity for the passive-on and the semi-active control schemes. Furthermore, the performance criterion J₂₀ indicates a drawback of the LQR(b) strategy. The scheme has satisfactory overall J₃ and J₁₁ performances, but requires a considerable number of sensors (41), while the SAC(a) strategy provides satisfactory J₃ and J₁₁ performance and robustness with the low requirement of 10 sensors.

Conclusion

In this study, an adaptive control approach is developed and designed aiming to mitigate seismic responses of bridges considering realistic implementation. The scheme is promising to control bridge structures given that it allows the choice of a model reference of significantly low order and it does not require full-state feedback or the use of observers. In addition, the adaptive technique is able to deal well with significant disturbances, and therefore it is promising to be utilized to mitigate seismic responses. Adaptive control is presented as an alternative to control bridge structures subjected to seismic excitation; as bridge structures are significant in size, there are many uncertainties involved in the estimation of bridges’ structural parameters, and bridges are susceptible to many external elements that may change their structural parameters significantly. The effectiveness of the control approach is investigated when controlling a seismically excited two-span highway bridge considering systematic parametric variations. The performance of the semi-active adaptive control scheme is compared to those of the semi-active non-adaptive and passive control schemes when the structure is subjected to a set of 11 earthquakes and to systematic parametric changes.

Results indicate that semi-active control is a more suitable alternative when controlling structures that have significant control placement constraints and are expected to have parametric variations. Passive schemes do not sustain performance and may increase midspan responses significantly. The optimal semi-active scheme reduces overall displacements as long as the requirement of observability is fulfilled. For large structures, as is the case for bridges, this requirement leads to a considerable number of sensors, thus increasing the installation, operation, and maintenance costs. The optimal strategy also shows potential to increase accelerations. The proposed semi-active adaptive controller with SAC holds performance well in face of parametric changes, is able to reduce overall seismic response, and requires a reasonable number of sensors. The approach is shown to be a promising and cost-effective alternative to control bridge structures, given that the estimation of structural parameters of bridges involves many uncertainties and they are prone to significant parametric variations over their service life.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Brazilian National Council for Scientific and Technological Development (CNPq; No. 203078/2014-4).

ORCID iDs

Rachel W Soares

Luciana R Barroso

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